Let $k$ be a nonarchimedian local field, $\widetilde{G}$ a connected reductive $k$-group, $\Gamma$ a finite group of automorphisms of $\widetilde{G}$, and $G:= (\widetilde{G}^\Gamma)^\circ$ the connected part of the group of $\Gamma$-fixed points of $\widetilde{G}$. The first half of my talk will concern motivation: a desire for a more explicit understanding of base change and other liftings of representations. Toward this end, we adapt some results of Kaletha-Prasad-Yu. Namely, if one assumes that the residual characteristic of $k$ does not divide the order of $\Gamma$, then they show, roughly speaking, that $G$ is reductive, the building $\mathcal{B}(G)$ of $G$ embeds in the set of $\Gamma$-fixed points of $\mathcal{B}(\widetilde{G})$, and similarly for reductive quotients of parahoric subgroups.

We prove similar statements, but under a different hypothesis on $\Gamma$. Our hypothesis does not imply that of K-P-Y, nor vice versa. I will include some comments on how to resolve such a totally unacceptable situation.

(This is joint work with Joshua Lansky and Loren Spice.)