A Study of Sapanisertib in Relapsed/Refractory NFE2L2-Mutated and Wild-Type Squamous Non-Small Cell Lung Cancer

Inclusion Criteria: * Stage IV squamous NSCLC. * Disease progression during or after prior systemic therapy for metastatic disease, which must include platinum-doublet chemotherapy and immune checkpoint inhibitor therapy (anti-PD-(L)1 +/-anti-CTLA-4), if approved and available, administered as separate lines of therapy or in combination. * Has study-eligible mutation in NFE2L2 or wild-type NFE2L2 using NGS from a College of American Pathologists- (CAP)-accredited and/or a Clinical Laboratory Improvement Amendments- (CLIA)-certified laboratory * Must have at least one radiographically measurable lesion per RECIST v1.1 defined as a lesion that is ≥ 10 mm in longest diameter or lymph node that is ≥ 15 mm in short axis imaged by computerized tomography (CT) scan or Magnetic Resonance Imaging (MRI). * Target lesions situated in a previously irradiated area may be considered measurable if progression has been demonstrated subsequent to radiation therapy. * Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-1 * Adequate Organ Function Laboratory Findings: Absolute neutrophil count (ANC): ≥1,500/mm3, Hemoglobin: ≥9.0 g/dL \* Transfusions and growth factors must not be used within 2 weeks prior to randomization to meet these requirements, Platelets: ≥ 100,000/mm3, Calculated creatinine clearance (CrCl): ≥ 40mL/min, Serum total bilirubin: ≤ 1.5× upper limit of normal (ULN) OR ≤ 3 mg/dL for patients with Gilbert's disease, AST (SGOT) and ALT (SGPT): ≤ 2.5× ULN OR ≤ 5× ULN for patients with liver metastases, Fasting triglycerides: \< 300 mg/dL, Fasting serum glucose: \<160 mg/dL * A female patient of childbearing potential must: 1. Have a negative serum or urine pregnancy test within 7 days prior to the first dose of study treatment 2. Agree to use acceptable methods of contraception(See Section 8.1.2) during the study and for a minimum of 14 days following the last dose of sapanisertib 3. Post-menopausal females (no menses for \>1 year without an alternative medical cause) and surgically sterilized females are exempt from these requirements. * Male patients must use an effective barrier method of contraception if sexually active with a female of childbearing potential and refrain from donating sperm during the study and for a minimum of 14 days following the last dose of sapanisertib. Exclusion Criteria: * Non-squamous cell histology and mixed histology tumors with any small-cell/neuroendocrine component. * Prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessment per investigator's discretion. * Receipt before the first dose of study drug of any of the following: i. Any investigational agent within 4 weeks. ii. Chemotherapy with 3 weeks (6 weeks for nitrosoureas or mitomycin C) iii. Any radiotherapy within 2 weeks prior to randomization with the exception of palliative radiotherapy for isolated tumor lesions * Major surgery or other anticancer therapy not previously specified within 4 weeks. * Unable or unwilling to discontinue proton pump inhibitor (PPI) use ≥ 5 days prior to randomization. * Interstitial lung disease or a history of pneumonitis that required oral or intravenous glucocorticoid treatment. * Any condition including social, psychiatric or medical (including uncontrolled significant concurrent illness) that in the opinion of the Investigator could interfere with treatment or protocol-related procedures. * Patients who are pregnant or lactating. * Symptomatic ascites or pleural effusion. Exception: Patients who are clinically stable following treatment for these conditions (including therapeutic thoraco-or paracentesis) are eligible. * Refractory nausea and vomiting, uncontrolled diarrhea, malabsorption, significant small bowel resection or gastric bypass surgery, use of feeding tubes or other situation that may preclude adequate absorption of oral study medication. * Infection requiring more than 5 days of parenteral antibiotics, antivirals, or antifungals within two weeks prior to randomization. * Patients receiving systemic corticosteroids greater than prednisone 10 mg or equivalent (excluding inhalers or low-dose hormone replacement therapy) within the 7 days before treatment initiation. * Previous intolerance to mammalian target of rapamycin (mTOR), AKT, or dual PI3K/mTOR inhibitors. * Patients with symptomatic, active/untreated central nervous system metastasis and/or leptomeningeal disease are not eligible. * Significant active cardiovascular disease * Participants who are known to be HIV-positive, unless assessed to be healthy with a low risk of AIDS-related outcomes. * Known active Hepatitis B or C infection. * Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption of the study drug.