Let $G=H\times J$, where $H\cong J\cong C\_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group cohomologies $H^\*(G,M)$?"


I tried to compute them via the Hochschild-Serre spectral sequence $E\_2^{p,q}=H^p(J,H^q(H,M))$. The computation of each term is easy, but I don't have any information about arrows and cannot determine $H^\*(G,M)$. 


More generally, let $G=C\_k\rtimes C\_2$ be a dihedral group and $M \cong \mathbb{Z}$ a $G$-module via "trivial $C\_k$-action and negation $C\_2$-action". Can one compute the group cohomologies $H^\*(G,M)$? 


I would appreciate it if you could provide an explicit computation, or a reference where I can find one. Thank you very much in advance.



A question on some computation of group cohomologies
See [Kuenneth-formula for group cohomology with nontrivial action on the coefficient](https://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient)


Let $C=C\_2$ be the cyclic group of order two, $\def\ZZ{\mathbb Z}\ZZ$ the trivial module over $C$ and $S$ the $C$-module which is $\ZZ$ as an abelian group with with the non-trivial action of $C$. You want to compute $H^\bullet(C\times C,M)$ with $M=\ZZ\otimes S$, and the Künneth formula tells us that there is a short exact sequence $$0\to (H^\bullet(C,\ZZ)\otimes H^\bullet(C,S))^p\to H^p(C\times C,M)\to (Tor^\ZZ\_1(H^\bullet(C,\ZZ),H^\bullet(C,S))^{p+1}\to0$$


You can easily compute $H^\bullet(C,-)$ with coefficients in every $C$-module and the Tor is also easy to compute. If I am not making too many mistakes, then \begin{gather}
(H^\bullet(C,\ZZ)\otimes H^\bullet(C,S))^{p}=(Tor\_1(H^\bullet(C,\ZZ),H^\bullet(C,S))^{p}=0
\end{gather}
for all even $p\geq0$, and
\begin{gather}
(H^\bullet(C,\ZZ)\otimes H^\bullet(C,S))^{p}=(\ZZ/2\ZZ)^r,\\
(Tor\_1(H^\bullet(C,\ZZ),H^\bullet(C,S))^{p}=(\ZZ/2\ZZ)^{r-1}
\end{gather}
for all odd $p=2r-1\geq0$. If follows that in the short exact sequenc above exactly one of the first and third terms are not zero for a given $p$, so we get an isomorphism in all cases.




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For the crossed product $G=C\_k\rtimes C\_2$ with coefficients $S=\mathbb Z$ with $C\_k$ acting trivially and $C\_2$ changing sizes. Notice that if $k$ is odd, then the H-L-S spectral sequence has second page $H^p(C\_2,H^q(C\_k,S))$, and this is zero except when $q=0$ and $p$ is odd, so this gives us the desired result in this case.


