Counting periodic points over finite fields

Let $V$ be a quasiprojective variety defined over ${\mathbb{F}}_q$, and let $\varphi :V\to V$ be an endomorphism of $V$ that is also defined over ${\mathbb{F}}_q$. Let $G$ be a finite subgroup of ${\mathrm{Aut}}_{{\mathbb{F}}_q}(V)$ with the property that $\varphi$ 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 $\varphi$ 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 $V\mathrm{/}G$ induced by $\varphi$ are related to periodic point counts on $V$ and all of its twists by $G$.