Counting periodic points on quotient varieties over Fq

Let V be a quasiprojective variety defined over Fq, and let ϕ:V→V be an endomorphism of V that is also defined over Fq. Let G be a finite subgroup of AutFq(V) with the property that ϕ commutes with every element of G. We show that idempotent relations in the group ring Q[G] give relations between the periodic point counts for the maps induced by ϕ on quotients of V by various subgroups of G. We also show that periodic point counts for the endomorphism on V/G induced by ϕ are related to periodic point counts on V and all of its twists by G.