  Using the functoriality of C. McCrory and A. Parusi\'nski's weight complex
-which induces an analog of the weight filtration for complex algebraic
varieties on the Borel-Moore homology with $\mathbb{Z}_2$ coefficients of real
algebraic varieties-, we define a weight complex with action on the real
algebraic varieties equipped with a finite group action. Emphasizing on the two
elements group, we then establish a filtered version of Smith short sequence,
taking into account the Nash-constructible filtration which realizes the weight
complex with action. Its exactness is implied by the splitting of a Nash
manifold equipped with an algebraic involution along an arc-symmetric subset.
