Let $G$ be an abelian group and $M$ be a $G$-module. The group of complex valued characters of $M$ is denoted by $\widehat M$. There is an obvious $G$-module structure on $\widehat M$.
Is there any relation between group cohomologies $H^n(G,M)$ and $H^n(G,\widehat M)$?
$$H^n(G,\widehat{M})\cong {H_n(G,M)}^{\widehat{\hphantom{u}}}$$ This is Proposition IV.7.1 in Brown's Cohomology of Groups. This comes from a pairing $$H^n(G,\widehat M)\times H_n(G,M)\to\Bbb Q/\Bbb Z.$$