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I came across the following Exercise in Brown's book "Co-homology of Groups", and have been completely unable to solve it. If anyone could give me a hint that would point me in the right direction I would really appreciate it. I couldn't even think of a filtration of $X$ which might produce a useful spectral sequence.

Brown "Co-homology of Groups", Section VIII.7, p.209, Ex 4:

Prove the following generalisation of 7.5: Let $X$ be a contractible $G$-CW-complex with finite isotropy groups and only finitely many cells mod-$G$. Then $H^*(G, \mathbb{Z} G) \cong H_c^*(X, \mathbb{Z})$.

I presume that he means $H^*(G, \mathbb{Z} G) \cong H_c^*(X, \mathbb{Z})$ as $\mathbb{Z}G$-modules.

There is a hint:

First show, by a spectral sequence argument for instance, that $H^*(G, \mathbb{Z}G)$ can be computed from $\text{Hom}_G(C(X), \mathbb{Z}G)$.

Here $H_c^*$ is the co-homology with compact supports. Throughout Brown's book any action on a space is always rigid (set-wise and point-wise stabilisers of cells coincide). The "7.5" which he refers to is the statement that $H^*(G, \mathbb{Z}G) \cong H_c^*(X, \mathbb{Z})$ as $\mathbb{Z}G$ modules, where $X$ is a contractible, free, co-compact $G$-CW-complex.

Edit: Having got no response here, I have cross-posted this to

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