I would like to know if there are explicit formulas for the Hodge-Deligne structure or the Hodge-Deligne polynomials for quotients X/G for finite groups G acting on a (smooth, projective) scheme X.

The only formula that I know is lemma 2.6 on this paper

http://arXiv.org/abs/math/0701642v1

by Munoz, Ortega and Vazquez-Gallo.