Known computations of certain 2-cohomology groups?

Question

I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$.

Here $A^{(\Gamma)}$ is the direct sum of copies of $A$ (one for each $g\in\Gamma$) and $\Gamma$ acts on $A^{(\Gamma)}$ via $(g.\xi)_h = \xi_{h g}$ for $\xi\in A^{(\Gamma)}$ and $g,h\in\Gamma$. Moreover, the cohomology theory under consideration is the (classical Eilenberg-MacLane) group cohomology of $\Gamma$ with coefficients in $A^{(\Gamma)}$.

Answer 1

When $A$ is a commutative ring, the module $A^{(\Gamma)}$ you describe coincides with the group ring $A[\Gamma]$ with it's canonical (right) $\Gamma$-action.

In particular, let $A=\mathbb{Z}$. Then $\Gamma$ is a duality group if there exists an integer $n$ such that $H^i(\Gamma ; \mathbb{Z}[\Gamma])=0$ for $i\neq n$ and $H^n(\Gamma;\mathbb{Z}[\Gamma])$ is torsion-free. In this case, $D=H^n(\Gamma;\mathbb{Z}[\Gamma])$ is called the dualizing module, and if $D \cong \mathbb{Z}$ then $\Gamma$ is called a Poincaré duality group. See Brown's "Cohomology of groups", section VIII.10, or Bieri and Eckmann's Inventiones paper "Groups with homological duality generalizing Poincaré duality"