Let $X$ be a nonsingular projective variety of dimension $n$ over a field $k$, and $\omega_X$ be its canonical sheaf. Let $G$ be a finite subgroup of the automorphism group $Aut_k(X)$, and $\mathcal{F}$ a locally free $G$-equivariant sheaf on $X$. Then $G$ acts on all the cohomology groups $H^i(X, \mathcal{F})$. Is the Serre duality $$ H^i(X, \mathcal{F})\times H^{n-i}(X, \mathcal{F}^\vee\otimes \omega_X)\to H^n(X, \omega_X)=k$$ a $G$-equivariant perfect pairing? Where can I find a reference to this result?

Thank you.