Let $X$ be a variety over $\mathbb{F}_q$. Let $G$ be a finite group of automorphisms of $X$ such that the map $f:X \to X/G$ is Galois. It is then the case that $H^i(X/G)=H^i(X)^G$ (where $H^i$ denotes étale cohomology with coefficients in $\mathbb{Q}_l$). As I understand, this follows from the Hochschild-Serre spectral sequence which can be found in the book by Milne.
Is there a reference which states the above explicitly (i.e. the isomorphism)? Moreover, is there a reference which shows that the above isomorphism is Frobenius equivariant? Or, is there a reference for the Frobenius equivariance of the Hochschild-Serre spectral sequence?