A spectral sequence for homology of invariant group chains

Let $Q$ be a finite group acting on a group $G$ by group automorphisms, $C(G)$ the bar complex and $H^Q_*(G,A)$ the homology of invariant group chains defined in K. Knudson's paper “The homology of invariant group chains”. In this paper we construct a spectral sequence converging to $H_*(Q,C(G)\otimes A)$ whose second term is isomorphic to $H^Q_*(G,A)$ for some coefficients. When this spectral sequence collapses this yields an isomorphism $H^Q_*(G,A)\cong H_*(Q,C(G)\otimes A)$, which we use to compute this homology for some cases. The construction uses a decomposition of the bar complex $C_*(G) $ in terms of the induction from some isotropy groups to the group $Q$. We also decompose the subcomplex of invariants $C_*(G)^Q$ by $Q$-orbits and use this to compute the invariant $1$-homology $H^Q_1(G,\mathbb Z)$ for some cases.