The quasi-polynomiality of mod q permutation representations for a linear finite group action on a lattice

For given linear action of a finite group on a lattice and a positive integer q , we prove that the mod q permutation representation is a quasi-polynomial in q . Additionally, we establish several results that can be considered as mod q -analogues of results by Stapledon for equivariant Ehrhart quasi-polynomials. We also prove a reciprocity-type result for multiplicities of irreducible decompositions.