Cobimetinib and Atezolizumab in Treating Participants With Advanced or Refractory Rare Tumors

Inclusion Criteria: * Patients must be informed of the investigational nature of this study and must be willing to give written informed consent in accordance with institutional and federal guidelines. Patients must be able to comply with the requirements and assessments of the study protocol * Must have histologically or cytologically documented rare tumor as defined per protocol that is metastatic or locally advanced and unresectable. Patients with locally advanced cutaneous squamous cell carcinoma that are technically resectable but in whom surgery is expected to lead to substantial function impairment or disfigurement are eligible * Must be refractory or intolerant to standard lines of therapy * Must have completed prior chemotherapy, immunotherapy, or radiation therapy at least 14 days prior to start of treatment and all toxicity must be resolved to Common Terminology Criteria for Adverse Events (CTCAE) v4.0 grade 1 (with the exception of CTCAE v4.0 grade 2 neuropathy) prior to start of treatment * Presence of radiographically evaluable disease * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Tissue Parameters: a. Representative formalin-fixed paraffin-embedded (FFPE) tumor specimens in paraffin blocks (blocks are preferred) or at least 4 unstained slides, with an associated pathology report, for testing of tumor PD-L1 expression (tumor tissue from bone metastases is not evaluable for PD-L1 expression and is therefore not acceptable). b. Tumor t…