Counting periodic points over finite fields

Let $V$ be a quasiprojective variety defined over $\mathbb{F}_q$, and let $\phi:V\rightarrow V$ be an endomorphism of $V$ that is also defined over $\mathbb{F}_q$. Let $G$ be a finite subgroup of $\operatorname{Aut}_{\mathbb{F}_q}(V)$ with the property that $\phi$ commutes with every element of $G$. We show that idempotent relations in the group ring $\mathbb{Q}[G]$ give relations between the periodic point counts for the maps induced by $\phi$ on the quotients of $V$ by the various subgroups of $G$. We also show that if $G$ is abelian, periodic point counts for the endomorphism on $V/G$ induced by $\phi$ are related to periodic point counts on $V$ and all of its twists by $G$.