Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet, Schüurmann, and Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson $L$-classes). In this paper we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds. © 2012 Wiley Periodicals, Inc.