Generic cohomology for twisted groups

Let $G$ be a simple algebraic group defined and split over ${k_0} = {{\mathbf{F}}_p}$, and let $\sigma$ be a surjective endomorphism of $G$ with finite fixed-point set ${G_\sigma }$. We give conditions under which cohomology groups of $G$ are isomorphic to cohomology groups of ${G_\sigma }$.