The homology of invariant group chains

If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_{\ast }(G{)}^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_{\ast }(G)$. In this paper, we study the homology of the complex $C_{\ast }(G{)}^Q$ and compute several examples.