  We first show the existence of a weight filtration on the equivariant
cohomology of real algebraic varieties equipped with the action of a finite
group, by applying group cohomology to the dual geometric filtration. We then
prove the compatibility of the equivariant weight filtrations and spectral
sequences with K{\"u}nneth isomorphism, cup and cap products, from the filtered
chain level. We finally induce the usual formulae for the equivariant cup and
cap products from their analogs on the non-equivariant weight spectral
sequences.
