Let $G=<\sigma>$ be a cyclic group of order $n$. For any $\mathbb{Z}[G]$ module $M$ it is known that the group cohomology
$$ H^i(G, M) = \begin{cases} M^G &\text{ if } i = 0 \\ M^G/NM &\text{ if } i \text{ odd} \\ ker(N)/(\sigma -1)M &\text{ if $i$ even} \end{cases}$$
where $N$ is the norm map appearing in the free resolution of $\mathbb{Z}$. I am wondering if there is a similar description for the group $G \times G$ and any $\mathbb{Z}[G\times G]$ module $M$.
In particular, if $G$ is a cyclic group of order $2$, $H^i(G,M) = M^G/NM$ for all $i>0$. I am wondering if in the case of $H^*(G\times G, M)$ the cohomology is also a quotient of $H^*(G; k) \otimes_k M^{G\times G}$ where $M$ is now a $k[G\times G]$ module and $k$ is a field with two elements.