Let $K$ be an algebraically closed field and $G$ a group.

Given a dg-algebra $A$, a left $A$-module $M$ and a right $A$-module $N$ let $Tor_A(M,N)$ denote the homology of the derived tensor product $M \otimes^L_A N,$ i.e. the realization of the Bar-construction $B(M,A,N) $ with $B(M,A,N)_n = M \otimes_K A^{\otimes n} \otimes N$.

There is a canonical map $\alpha: K \otimes^L_{C^*(BG;K)} K \to C^*(G;K) $ of dg-algebras over $K$ (in fact of $E_\infty$-algebras over $K$) that induces on homology a graded map $$ \beta: Tor_{C^*(BG;K)}(K,K) \to K^G. $$

What can we say about the map $\beta$ if $G$ is a derived $p$-complete abelian group, say for example $\mathbb{Z}^\wedge_p$ or $\mathbb{Z}/p^n \mathbb{Z}?$