Treatment With Amivantamab and Hyaluronidase or Cetuximab for Advanced Skin Cancer in People With a Weakened Immune System

PRIMARY OBJECTIVES: I. To evaluate the safety and preliminary efficacy of amivantamab monotherapy in patients with locoregionally incurable or metastatic cutaneous squamous cell carcinoma and an active immunosuppressed condition. (Cohort A) II. To compare progression-free survival (PFS) of amivantamab monotherapy versus cetuximab monotherapy in patients with locoregionally incurable or metastatic cutaneous squamous cell carcinoma and an active immunosuppressed condition. (Cohort B) SECONDARY OBJECTIVES: I. To estimate the frequency and severity of toxicities in each cohort and treatment arm. II. To estimate confirmed objective response rate (ORR) (ORR, confirmed complete and partial responses by Response Evaluation Criteria in Solid Tumors \[RECIST\] version \[v\] 1.1) in each cohort and treatment arm. III. To estimate duration of objective response, time to progression, time to next treatment, and overall survival in each cohort and treatment arm. IV. To estimate ORR and progression free survival (PFS) in each cohort and treatment arm in subgroups defined by reason for immunosuppression: transplant, autoimmune disease, hematologic malignancy, or other reason. BANKING OBJECTIVE: I. To bank specimens for future correlative studies. OUTLINE: Patients in cohort A are assigned to arm I, patients in cohort B are randomized to arm I or II. ARM I: Patients receive amivantamab and hyaluronidase subcutaneously (SC) over at least 5 minutes on days 1, 8, 15 and 22 of cycle 1 and day 1 of subsequent cycles. Cycles repeat every 28 days for 24 cycles in the absence of disease progression or unacceptable toxicity. Patients undergo blood sample collection on study and computed tomography (CT) and/or magnetic resonance imaging (MRI) throughout the study. ARM II: Patients receive cetuximab intravenously (IV) on days 1 and 15 of each cycle. Cycles repeat every 28 days for 24 cycles in the absence of disease progression or unacceptable toxicity. Patients undergo blood sample collection on study and CT and/or MRI throughout the study. After completion of study treatment, patients are followed up, per the treating investigator, for up 3 years.