Testing of TAK228 (MLN0128, Sapanisertib) Plus Docetaxel to the Usual Standard of Care for Advanced Squamous Cell Lung Cancer (A Lung-MAP Treatment Trial)

Inclusion Criteria: * Patients must be assigned to S1900D. Assignment to S1900D is determined by the S1400 or LUNGMAP protocol genomic profiling using the FoundationOne assay. Biomarker eligibility for S1900D is based on the identification of an NFE2L2 mutation or KEAP1 alteration * Patients must have a histologically or cytologically confirmed stage IV or recurrent pure squamous cell lung cancer * Patients must have measurable disease documented by computed tomography (CT) or magnetic resonance imaging (MRI). The CT from a combined positron emission tomography (PET)/CT may be used to document only non-measurable disease unless it is of diagnostic quality. Measurable disease must be assessed within 28 days prior to sub-study randomization. Pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease. Non-measurable disease must be assessed within 42 days prior to sub-study randomization. All disease must be assessed and documented on the Baseline Tumor Assessment Form. Patients whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to randomization. CT and MRI scans must be submitted for central review via transfer of images and data (TRIAD) * Patients must have a CT or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to sub-study randomization * Patients with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load on suppressive therapy within 28 days prior to sub-study randomization * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. Patients with HCV infection who are currently on treatment must have an undetectable HCV viral load within 28 days prior to sub-study randomization * Patients with a known history of human immunodeficiency virus (HIV) seropositivity: * Must have undetectable viral load using standard HIV assays in clinical practice * Must have CD4 count \>= 400/mcL * Must not require prophylaxis for any opportunistic infections (i.e., fungal, mycobacterium avium complex \[MAC\], or pneumocystis pneumonia \[PCP\] prophylaxis) * Must not be newly diagnosed within 12 months prior to sub-study randomization * Patients must be able to swallow oral medications. Patients must not have any impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of TAK228 (MLN0128, sapanisertib) (e.g. ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection, or enteric stomata) * Patients must be able to safely receive at least one of the investigator's choice of standard of care regimens * Patients must have progressed (in the opinion of the treating physician) following the most recent line of therapy * Patients must have recovered (=\< grade 1) from any side effects of prior therapy * Absolute neutrophil count (ANC) \>= 1,500/mcl (obtained within 28 days prior to sub-study randomization) * Platelet count \>= 100,000 mcl (obtained within 28 days prior to sub-study randomization) * Hemoglobin \>= 9 g/dL (obtained within 28 days prior to sub-study randomization). Transfusion to obtain a hemoglobin (Hb) of \>= 9 g/dl is acceptable * Patients must have fasting triglycerides =\< 300 mg/dL within 28 days prior to sub-study randomization * Serum bilirubin =\< Institutional upper limit of normal (IULN) (within 28 days prior to sub-study randomization. For patients with liver metastases, serum bilirubin must be =\< 2 x IULN * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) =\< 2 x IULN (within 28 days prior to sub-study randomization) (if both ALT and AST are done, both must be \< 2 IULN). For patients with liver metastases, ALT or AST must be =\< 5 x IULN (if both ALT and AST are done, both must be =\< 5 x IULN) * Patients must have a serum creatinine =\< the IULN or calculated creatinine clearance \>= 40 mL/min using the following Cockcroft-Gault formula. This specimen must have been drawn and processed within 28 days prior to sub-study randomization * Patients must have Zubrod performance status 0-1 documented within 28 days prior to sub-study randomization * Pre-study history and physical exam must be obtained within 28 days prior to randomization * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for two years. In addition, indolent or low-grade malignancies which are not expected to interfere with assessment of the primary or toxicity endpoints are permissible (e.g. localized well differentiated prostate cancer). These cases should be discussed with a study chair prior to enrollment * Patients must agree to have blood specimens submitted for circulating tumor DNA (ctDNA) * Patients must also be offered participation in banking and in the correlative studies for collection and future use of specimens * Patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * As a part of the OPEN registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system * Patients with impaired decision-making capacity are eligible as long as their neurological or psychological condition does not preclude their safe participation in the study (e.g., tracking pill consumption and reporting adverse events to the investigator) Exclusion Criteria: * Patients must not have EGFR sensitizing mutations, EGFR T790M mutation, ALK gene fusion, ROS 1 gene rearrangement, and BRAF V600E mutation unless they have progressed following all standard of care targeted therapy * Patient must not have leptomeningeal disease, spinal cord compression or brain metastases unless: (1) metastases have been locally treated and have remained clinically controlled and asymptomatic for at least 14 days following treatment, and prior to sub-study randomization, AND (2) patient has no residual neurological dysfunction and has been off corticosteroids for at least 14 days prior to sub-study randomization. Patients with brain metastases that have not been treated may be registered if the metastases are: * \< 2 mm OR deemed clinically equivocal, AND * No evidence of shift, AND * No focal or other neurologic deficits, AND * No requirement for steroids, anti-seizure medications * Patients must not have uncontrolled illnesses within 28 days prior to sub-study randomization including: * Uncontrolled asthma is defined as: oxygen (O2) saturation \< 90% by arterial blood gas analysis or pulse oximetry on room air * Medically significant (symptomatic) bradycardia * History of arrhythmia requiring an implantable cardiac defibrillator * Symptomatic pulmonary hypertension * Uncontrolled hypertension (use of anti-hypertensive agents to control hypertension before cycle 1 day 1 \[C1D1\] is allowed) * Clinically significant valvular disease * Patients must not have received any radiation therapy within 14 days prior to randomization * Patients must not be planning to receive any concurrent chemotherapy, immunotherapy, biologic or hormonal therapy for cancer treatment while receiving treatment on this study. Concurrent use of hormones for non-cancer-related conditions (e.g. insulin for diabetes and hormone replacement therapy) is acceptable * Patient must not have had a major surgery within 14 days prior to sub-study randomization. Patient must have fully recovered from the effects of prior surgery in the opinion of the treating investigator * Patients must not have previously received treatment with PI3K, AKT, PI3K/mTOR inhibitors, TORC1/2 inhibitors or TORC1 inhibitors * Patients must not have uncontrolled diabetes within 28 days prior to sub-study randomization * Definition of uncontrolled diabetes: Glycosylated hemoglobin (HbA1c) \>= 8.0% and fasting serum glucose (\> 130 mg/dL) within 28 days prior to sub-study randomization * Patients must not have any grade III/IV cardiac disease as defined by the New York Heart Association Criteria (i.e., patients with cardiac disease resulting in marked limitation of physical activity or resulting in inability to carry on any physical activity without discomfort), unstable angina pectoris, myocardial infarction within 6 months, or serious uncontrolled cardiac arrhythmia * Patients must not have had an ischemic cerebrovascular event or pulmonary embolism within 6 months * Patients must not have a rate-corrected QT interval \> 480 msec based on the 12-lead electrocardiography (ECG) within 28 days prior to sub-study randomization (i.e., repeated demonstration of corrected QT \[QTc\] interval \> 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes). It is suggested that a local cardiologist review the QTc intervals * Patients must not have received any live attenuated vaccinations or been in close contact with those who have received live vaccines within 28 days prior to sub-study registration and throughout protocol treatment * Patients must not be planning to use proton pump inhibitors (PPIs) at least one week prior to sub-study randomization and throughout protocol treatment * Patients must not be pregnant or nursing. Women of reproductive potential must have agreed to use an effective contraceptive method and must not donate eggs during protocol treatment and for 90 days after the last protocol treatment. Men of reproductive potential must have agreed to use an effective contraceptive method and must not donate sperm during protocol treatment and for 120 days after the last protocol treatment. A woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months. In addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation. However, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures