Let F be a finite group and X be a complex quasi-projective F-variety. For r \in N, we consider the mixed Hodge-Deligne polynomials of quotients Xr/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G=GL(n,C) and SL(n,C), we get even more concrete expressions for these polynomials, using the combinatorics of partitions.