For algebraic actions of finite groups on singular complex algebraic varieties, equivariant Hirzebruch characteristic classes have been defined by Cappell, Maxim, Schürmann and Shaneson. The corresponding picture for an equivariant Goresky-MacPherson L-class was elusive so far. We will discuss a direct homological construction of a purely topological equivariant theory for the L-class of pseudomanifolds. This involves in particular an analysis, akin to an "intersection homology Conner conjecture", of Poincaré duality properties of orbit spaces. In applications, the equivariant classes serve as a tool to compute the L-class of the orbit space. The K-theoretic perspective on this is joint work with Eric Leichtnam and Paolo Piazza.