Let $G$ be a finite group, let $kG$ denote the group ring over a field $k$, and let $P_*$ be a free resolution of $k$ as $kG$-module. If $A$ is a trivial $kG$-module. There is an isomorphism $$ P_* \otimes_ G A \cong (P_* \otimes_G k) \otimes A$$
By the universal coefficient theorem, there is an isomorphism
$$H_*(G,A) = H_*(P_* \otimes_G A) \cong H_*(P_* \otimes_G k) \otimes A = H^*(G) \otimes A$$
Similarly, if $A_*$ is a chain complex of trivial $kG$-modules, by the Kunneth theorem we have an isomorphism
$$H_*(G,A_*) \cong H_*(G) \otimes H_*(A_*)$$
Now I am interested in the dual case of cohomology. Namely, for a trivial $kG$-module $A$, there is an isomorphism
$$Hom_G(P_*,A) \cong Hom(P_* \otimes_G k, A)$$
and by the universal coefficient theorem we have that $H^*(G,A) \cong \hom(H_*(G), A)$ (Is the latter isomorphic to $H^*(G) \otimes A$ ?)
If $A^*$ is a cochain complex of trivial $kG$-modules, is it also true that $H^*(G;A) = H^*(G) \otimes H^*(A)$ ?