Neratinib and Everolimus, Palbociclib, or Trametinib in Treating Participants With Refractory and Advanced or Metastatic Solid Tumors With EGFR Mutation/Amplification, HER2 Mutation/Amplification, or HER3/4 Mutation or KRAS Mutation

Inclusion Criteria: 1. Subjects with advanced or metastatic solid tumors that are refractory to standard therapies known to provide clinical benefit. Subjects with hematologic malignancy including lymphoma/myeloma will not be enrolled on this study. 2. Subjects must have one of the following: 1. somatic mutations in human epidermal growth factor receptor (EGFR, HER2, HER3, and HER4) 2. EGFR gene amplification (patients with 3+ results on immunohistochemistry testing for EGFR may be allowed to enroll if gene amplification results are unavailable) 3. HER2 gene amplification (patients with 3+ results on immunohistochemistry testing for Her-2 may be allowed to enroll if gene amplification results are unavailable) 4. Somatic mutation in KRAS (Patients will be enrolled only on neratinib and trametinib combination ARM). 3. Subjects must have measurable disease by RECIST v1.1. (only for MTD Expansion Cohorts) 4. Subjects must be ≥18 years of age. 5. Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1. 6. Abnormal organ function is permitted. However, subjects must have: 1. absolute neutrophil count ≥ 1500/mL 2. platelets ≥ 100,000/mL 3. hemoglobin ≥ 9 g/dL 4. creatinine ≤ 1.5 X upper limit of normal (ULN) or CrCl \>40 ml/min 5. total bilirubin ≤ 1.5 X ULN 6. aspartate aminotransferase (AST/SGOT) and/or alanine aminotransferase (ALT/SGPT) ≤ 2.5 X ULN (≤5 X ULN in subjects with liver metastases) 7. Subjects must be ≥4 weeks or at least 5 half-lives beyond treatment with any chemotherapy or other investigational therapy to include hormonal, biological, or targeted agents, whichever is shorter at the time of treatment initiation. Subjects must be ≥4 weeks beyond treatment with wide field radiation therapy or ≥2 weeks for limited field radiation therapy for palliation (not including whole brain radiation or stereotactic radiotherapy, for which subjects must be ≥4 weeks beyond treatment). 8. Women of child-bearing potential MUST have a negative serum or urine human chorionic gonadotropin (HCG) test unless prior hysterectomy or menopause (defined as 12 consecutive months of amenorrhea). Subjects should not become pregnant or breastfeed while on this study. Sexually active subjects must agree to use contraception for the duration of study participation and for 4 months after the last dose of neratinib and everolimus, palbociclib or trametinib. 9. Ability to understand and willingness to sign informed consent form prior to initiation of the study and any study procedures. Only for subjects enrolled in Arm 1 - Neratinib and Everolimus 10. Fasting lipid profile: Cholesterol less than or equal to 350 mg/dL and triglycerides less than or equal to 400 mg/dL. 11. Subjects who are taking medications with potent inhibitors or inducers of CYP450 3A4 should be off for 5 half-lives prior to starting everolimus. Only for subjects enrolled in Arm 2 - Neratinib and Palcociclib 12. Any prior neuropathy should be back to baseline or grade 1 13. Subjects who are taking medications with potent inhibitors or inducers of CYP450 3A4 should be off for 5 half-lives prior to starting Palbociclib. Only for subjects enrolled in Arm 3 - Neratinib and Trametinib 14. All skin rash (dermatitis acneiform, erythema, xeroderma, eczema) should be at grade 1 when starting trametinib treatment. 15. History of retinal disorder, dry eye syndrome, or blurry vision need to be evaluated by ophthalmology prior to starting treatment. Exclusion Criteria: 1. Subjects who are pregnant or breastfeeding; 2. Known Hepatitis B, Hepatitis C or human immunodeficiency virus (HIV) infection. 3. Inability or unwillingness to swallow pills. 4. Active infection requiring intravenous (IV) antibiotics or other uncontrolled intercurrent illness requiring hospitalization. 5. Clinically significant gastrointestinal (GI) abnormalities that may alter absorption such as malabsorption syndrome or major resection of the stomach or bowels. For example, subjects should have no more than 50% of the large intestine removed and no sign of malabsorption (e.g. gastrectomy, ileal bypass, chronic diarrhea, Crohn's disease, malabsorption, gastroparesis). 6. Inability to comply with the study and follow-up procedures. 7. History of cerebrovascular accident (CVA), myocardial infarction or unstable angina within the previous 6 months before starting therapy. 8. Prolongation of QT/QTc interval (QTc interval \>450 ms for males or \>470 ms for females) using the Fredericia method of QTc analysis. For patients with bundle branch block (BBB), the Rautaharju method of QTc analysis (QTcR) will be used. 9. Has active primary brain tumor, active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are clinically stable for at least 4 weeks with no neurological symptoms, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless or clinical stability. 10. Uncontrolled concurrent disease or illness including but not limited to: * symptomatic congestive heart failure (NYHA Class III or IV) per the NYHA Classification (see Appendix B), unstable angina pectoris, clinically significant cardiac arrhythmia * unstable or untreated cardiac conditions or ejection fraction of \<50% as determined by echocardiogram (ECHO) or multiple gated acquisition scan (MUGA) * diabetes mellitus (i.e. fasting blood glucose \>220 despite acceptable chronic diabetes therapy) * psychiatric illness that would limit compliance with study requirements, as determined by the investigator 11. Participating in any other clinical trials using an investigational product. Only for subjects enrolled in Arm 1 - Neratinib and Everolimus 12. History of hypersensitivity to everolimus 13. Subjects requiring therapy with immunosuppressive agents such as anti-tumor necrosis factor alpha (TNFα) agents (Etanercept, Adalimumab), azathioprine, methotrexate, cyclosporine, etc for active autoimmune disorder. 14. Major surgery ≤28 days prior to treatment with everolimus. Only for subjects enrolled in Arm 3 - Neratinib and Trametinib 15. Albumin less than 3 Gm/dL