On a generalization of Fermat's little theorem

We obtain a congruence type arithmetic relation on the set of all triples $(G,H,P)$, where $G$ is a finite group, $H$ is a subgroup, and $P$ is a representation of $G$ by permutations. This relation becomes Fermat's Little Theorem in the case when $G=Z_p$, $H=1$, and $P$ is the regular representation of $G$.