We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of McCrory and Parusiński. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence.
Nevertheless, we construct finite additive invariants in terms of bounded long exact sequences, recovering Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these additive invariants by using globally invariant chains and the equivariant version of Guillén and Navarro Aznar's extension criterion.
Fabien PRIZIAC. "Equivariant weight filtration for real algebraic varieties with action." J. Math. Soc. Japan 68 (4) 1789 - 1818, October, 2016. https://doi.org/10.2969/jmsj/06841789