Trastuzumab + Alpelisib +/- Fulvestrant vs Trastuzumab + CT in Patients With PIK3CA Mutated Previously Treated HER2+ Advanced BrEasT Cancer (ALPHABET)

Inclusion Criteria: Patients are eligible to be enrolled for randomization in the study only if they meet all of the following criteria: 1. Written informed consent prior to any specific study procedures, showing patient willingness to comply with all study procedures. 2. Histologically or cytologically documented locally recurrent inoperable or metastatic breast cancer with HER2+ status based on local laboratory determination, preferably on the most recent available FFPE tumor sample, and according to American Society of ClinicalOncology (ASCO)/College of American Pathologists (CAP) international guidelines valid at the time of the assay. In case of discordance in HER2+ status in different biopsies, we will consider the result from the most recent biopsy one will be used. 3. Documented HR status based on local laboratory, preferably on the most recent available FFPE tumor sample, and according to ASCO/CAP international guidelines valid at the time of the assay. In case of discordance in HR status in different biopsies, the result from the most recent biopsy will be used. HR+ will be defined as ≥1% positive cells by immunohistochemistry for Estrogen Receptor (ER) and/or Progesterone Receptor (PgR). HR- will be defined as \<1% positive cells by immunohistochemistry for both ER and PgR. Considering that there are limited data on endocrine therapy benefit for cancers with 1% to 10% of cells staining ER positive, for the purpose of this study, patients with ER and PgR expression between 1 and 10% (considered to be HR low by the most recent ASCO/CAP guidelines) will be eligible for inclusion in the HR- cohort. 4. Patients with a PIK3CA tumor mutation at central laboratory determination, preferably on the most recent available FFPE tumor sample. 5. At least 1 but no more than 5 prior lines of anti-HER2 based therapy for metastatic breast cancer (MBC). Maintenance therapy will not count as an additional line of therapy. 6. At least 1 prior line of trastuzumab in the metastatic setting, or in the (neo)adjuvant setting (provided the patient relapsed while on therapy or within 6 months after completing adjuvant trastuzumab). 7. Female or male patient is at least 18 years of age. 8. Eastern Cooperative Oncology Group (ECOG) Performance Status 0-1. 9. Patients can be either males or premenopausal/perimenopausal or postmenopausal females. In the HR+ cohort, males and females who are not post-menopausal must have been on a gonadotropin-releasing hormone (GnRH) agonist (e.g. goserelin or leuprorelin) for at least 28 days prior to starting study treatment. Premenopausal status is defined as either: * Last menstrual period occurred within the last 12 months, or * If on tamoxifen: last menstrual period occurred within the past 14 days, plasma estradiol is ≥ 10 pg/mL and follicle-stimulating hormone (FSH) ≤ 40 IU/l or in the premenopausal range, according to local laboratory definition, or * In case of therapy induced amenorrhea: plasma estradiol is ≥ 10 pg/mL and FSH ≤ 40 IU/l or in the premenopausal range, according to local laboratory definition. Postmenopausal status is defined as either: \- Natural (spontaneous) amenorrhea lasting more than 12 months and either age from49 to 59 years and/or history of vasomotor symptoms (e.g., hot flush) in the absence of other medical justification, or Levels of plasma estradiol ≤ 20 pg/mL and follicle-stimulating hormone (FSH) ≥ 40 IU/l or in the postmenopausal range, according to local laboratory definition, or Surgical bilateral oophorectomy. Perimenopausal status is defined as neither premenopausal nor postmenopausal. 10. Measurable disease or at least one evaluable bone lesion, lytic or mixed (lytic+blastic), which has not been previously irradiated and is assessable by computer tomography (CT)/magnetic resonance imaging (MRI) in the absence of measurable disease according to RECIST 1.1 criteria. 11. Life expectancy ≥ 12 weeks. 12. Adequate organ and marrow function defined as follows: * Absolute neutrophil count (ANC) ≥ 1,500/mm3 (1.5x109/L). * Platelets ≥ 100,000/mm3 (100x109/L). * Hemoglobin ≥ 9g/dL (90g/L). * Calcium (corrected for serum albumin) and magnesium within normal limits or ≤ grade 1 according to NCI-CTCAE version 5.0 if judged clinically not significant by the investigator. * Creatinine \<1.5 x upper limit of normal (ULN) or creatinine Clearance ≥ 35 mL/min using Cockcroft-Gault formula (if creatinine is ≥1.5 ULN). * Total bilirubin \< 2 x ULN (any elevated bilirubin should be asymptomatic at enrollment) except for patients with Gilbert's syndrome who may only be included if the total bilirubin is ≤ 3.0 x ULN or direct bilirubin ≤ 1.5 x ULN. * Potassium within normal limits, or corrected with supplements. * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3.0 x ULN. If patient has liver metastasis, AST and ALT ≤ 5.0 x ULN (elevated AST or AST values must be stable for 2 weeks, without evidence of biliary obstruction by imaging). * Fasting serum amylase ≤ 2.0 x ULN. * Fasting serum lipase ≤ ULN. * Fasting plasma glucose (FPG) \< 140 mg/dL (7.7 mmol/L) and glycosylated hemoglobin (HbA1c) \< 6.5%. 13. Resolution of all acute toxic effects of prior anti-cancer therapy or surgical procedures to NCI-CTCAE version 5.0 grade ≤ 1 (except alopecia or other toxicities not considered a safety risk for the patient at investigator´s discretion). 14. Adequate cardiac function as defined by left ventricular ejection fraction (LVEF) of ≥ 50% measured by echocardiography or multi-gated acquisition (MUGA) scans. Exclusion Criteria: Patients will be excluded from the study if they meet any of the following criteria: 1. Have received more than 5 previous lines of anti-HER2 based therapy for MBC, or prior fulvestrant. 2. Symptomatic visceral disease or any disease burden that makes the patient ineligible for experimental therapy per the investigator's best judgment. 3. Symptomatic central nervous system (CNS) metastases. However, patients with CNS metastases who have been adequately treated, are asymptomatic and do not require corticosteroid or anti-epileptic medication are eligible. 4. Presence of leptomeningeal carcinomatosis. 5. Other invasive malignancy (different from the current breast cancer) at the time of enrollment or previous diagnosis of a completely removed malignancy within 3 years prior to randomization except for adequately treated (including complete surgical removal) of International Federation of Gynecology and Obstetrics (FIGO) stage I grade 1 endometrial cancer, basal or squamous cell carcinoma of the skin, thyroid cancer limited to thyroid gland, in situ carcinoma of the cervix, and grade 1-2 early stage bladder cancer defined as T1 or less, without nodal involvement (N0). 6. Patients with an established diagnosis of diabetes mellitus type I or not controlled type II (FPG ≥ 140 mg/dL \[7.7 mmol/L\] or HbA1c ≥ 6.5%), or history of gestational diabetes (as per American College of Obstetricians and Gynecologists (ACOG) guidelines) or documented steroid-induced diabetes mellitus. 7. Prior treatment with any mTOR, AKT or PI3K inhibitor. 8. Patients treated within the last 7 days prior to treatment initiation with: * Drugs that are strong inducers of CYP3A4. * Drugs that are inhibitors of Breast Cancer Resistance Protein (BCRP). 9. Patients who received before randomization: * Any investigational agent within 4 weeks. * Chemotherapy within a period of time that is shorter than the cycle duration used for that treatment (e.g. \< 3 weeks for fluorouracil, doxorubicine, epirubicin or \< 1 week for weekly chemotherapy). * Biologic therapy (e.g., antibodies, other than trastuzumab which is permitted): within 4 weeks prior to starting study treatment. * Endocrine therapy: tamoxifen or aromatase inhibitor (AI) within 2 weeks prior to starting study treatment. * Corticosteroids within 2 weeks prior to starting study treatment. Note: the following uses of corticosteroids are permitted at any time: single doses, topical applications(e.g., for rash), inhaled sprays (e.g., for obstructive airways diseases), eye drops or local injections (e.g., intra-articular). * Radiotherapy within 2 weeks prior to starting study treatment (all acute toxic effects must be resolved to NCI-CTCAE version 5.0 grade \<1, except toxicities not considered a safety risk for the patient at investigator´s discretion). Patients who received prior radiotherapy to \>25% of bone marrow are not eligible regardless of when it was administered. * Major surgery or other anti-cancer therapy not previously specified within 4 weeks prior to starting study treatment, (all acute toxic effects, including peripheral neurotoxicity must be resolved to NCI-CTCAE version 5.0 grade ≤ 1, except toxicities not considered a safety risk for the patient at the investigator´s discretion). 10. Patient has clinically significant, uncontrolled heart disease and/or recent cardiac events including any of the following: * History of angina pectoris, coronary artery bypass graft (CABG), symptomatic pericarditis or myocardial infarction within 6 months of randomization. * History of documented congestive heart failure (New York Heart Association functional classification III-IV). * Clinically significant cardiac arrhythmias, (e.g., ventricular tachycardia), complete left bundle branch block, high grade AV block (e.g. bifascicular block, Mobitz type II and third degree AV block without pacemaker in place). * Uncontrolled hypertension defined by a Systolic Blood Pressure ≥ 160 mmHg and/or Diastolic Blood Pressure (DBP) ≥ 100 mm Hg, with or without anti-hypertensive medication. Initiation or adjustment of antihypertensive medication(s) is allowed prior to screening. * Long QT syndrome, family history of idiopathic sudden death or congenital long QT syndrome, or Fridericia QT correction formula (QTcF) \> 470msec. * Bradycardia (heart rate \< 50 at rest), by electrocardiogram (ECG) or pulse. * Inability to determine the QTcF interval on the ECG (i.e.: unreadable or not interpretable) or QTcF \> 460 msec for females (using Fridericia's correction). All as determined by screening ECG. 11. Bleeding diathesis (i.e., Disseminated Intravascular Coagulation (DIC), clotting factor deficiency) or long-term (\> 6 months) anticoagulant therapy, other than antiplatelet therapy and low dose coumarin derivatives, provided that the International Normalised Ratio (INR) is less than 1.5. 12. History of clinically significant bowel disease including abdominal fistula, or gastrointestinal perforation. 13. Difficulties to swallow tablets, malabsorption syndrome disease significantly affecting gastrointestinal function, resection of the stomach or small bowel, or active inflammatory bowel disease (e.g., ulcerative diseases). 14. Known hypersensitivity to trastuzumab, alpelisib or fulvestrant or any of their excipients. If known hypersensitivity to either vinorelbine, capecitabine, eribulin or any of their excipients, patient will be eligible as long as the investigator's choice avoids that drug in the control arm. If known hypersensitivity to all three cytostatics (vinorelbine, capecitabine and eribulin), the patient will not be eligible. 15. Known positive serology for Human Immunodeficiency Virus (HIV), or active infection for hepatitis B or hepatitis C. 16. Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or study treatment administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the patient inappropriate for entry into this study. 17. Patients with currently documented pneumonitis/interstitial lung disease (the chest Computed Tomography \[CT\] scan performed at screening for the purpose of tumor assessment should be reviewed to confirm that there are no relevant pulmonary complications present). 18. Patient with liver disease with a Child Pugh score B or C. 19. Patient with a history of acute pancreatitis within 1 year of screening or past medical history of chronic pancreatitis. 20. Patient has a history of Steven-Johnson-Syndrome (SJS) or Toxic Epidermal Necrolysis (TEN). 21. Patient is nursing (lactating) or is pregnant as confirmed by a positive serum human Chorionic Gonadotropin (hCG) test prior to initiating study treatment. 22. Patient is a woman of child-bearing potential or a partner of a woman of child-bearing potential, unless agreement to remain abstinent or use single or combined non-hormonal contraceptive methods that result in a failure rate of \< 1% per year during the treatment period and for at least 7 months after the last dose of study treatment, except for patients receiving fulvestrant in which this period should be of at least 2 years. * Abstinence is only acceptable if it is in line with the preferred and usual lifestyle of the patient. * Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception. * Examples of non-hormonal contraceptive methods with a failure rate of \< 1% per year include tubal ligation, male sterilization (only if he is the sole partner and have been performed at least 6 months prior to screening), and certain intrauterine devices. * Alternatively, two methods (e.g., two barrier methods such as a condom and a cervical cap) may be combined to achieve a failure rate of \< 1% per year. Barrier methods must always be supplemented with the use of a spermicide. * Male participants must not donate sperm during study and up to the time period specified above.