Necitumumab and Trastuzumab in Combination With Osimertinib for the Treatment of Refractory Epidermal Growth Factor Receptor (EGFR)-Mutated Stage IV Non-small Cell Lung Cancer

Inclusion Criteria: * Willing and able to provide informed consent * Cytologically or histologically confirmed non-small cell lung cancer (NSCLC), that is stage IV (metastatic), with an activating and sensitizing EGFR mutation (e.g., exon 20 insertion mutations are excluded). Enrollment of patients with mutations other than exon 19 deletion and the L858R point mutation require literature supporting sensitivity to EGFR tyrosine kinase inhibitors * Progressed on osimertinib. Osimertinib must have been included as the last systemic therapy prior to trial enrollment. This excludes patients who received osimertinib in combination with other EGFR-tyrosine kinase inhibitor (TKI) or anti-human epidermal growth factor receptor (anti-HER) therapy * Measurable disease, as per RECIST 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Able to swallow the study drugs, has no known intolerance of study drugs or excipients, and able to comply with study requirements * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L * Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance \>= 50 mL/min for participants with creatinine levels \> 1.5 X institutional ULN (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) * Creatinine clearance may be calculated following institutional practices * Serum total bili…