  Given a $p$-adic group $G$ equipped with an action of a finite group
$\Gamma\subset\mathrm{Aut}_F(\mathbf{G})$, and a reductive fixed-point subgroup
$G^\Gamma$, we establish a relationship between constructions of types for
these two groups due to Yu, Kim--Yu and Fintzen, and generalizes the
relationship between general linear and classical groups observed by Stevens.
As an application, given a parabolically induced or supercuspidal
representation $\pi$ of $G$, we explicitly identify a number of the inertial
equivalence classes occurring in the representation $\pi|_{G^{[\Gamma]}}$.
