Functoriality, Smith theory, and the Brauer homomorphism

If $\sigma$ is an automorphism of order $p$ of the semisimple group $\mathbf{G}$, there is a natural correspondence between mod $p$ cohomological automorphic forms on $\mathbf{G}$ and ${\mathbf{G}}^{\sigma }$. We describe this correspondence in the global and local settings.