Restricting admissible representations to fixed-point subgroups

Given a p-adic group G=G(F) and finite Γ⊂AutF(G) such that the fixed-point subgroup GΓ is reductive, we establish relationship between constructions of types for G due to Yu, Kim–Yu Fintzen, which generalizes those general linear classical groups originally observed by Stevens. As an application, given irreducible representation π G, explicitly identify number inertial equivalence classes occurring in π|G[Γ].