Suppose $G$, $H$ are finite groups and $M$ is a module over $G\times H$. 


**Question:** Is the exponent of $H^i(G\times H,M)$ a divisor of $lcm(|G|,|H|)$ for $i> 0$ ? 


The Künneth formula answers the question affirmatively if $M$ is trivial or, more generally, if one of the groups acts trivially on $M$. But I don't know what to expect if $M$ is non-trivial. 



Exponent of the cohomology of a product of groups
For any finite group $\Gamma$, if $I$ is the augmentation ideal of ${\mathbb Z}\Gamma$, then $H^1(\Gamma,I)\cong {\mathbb Z}/|\Gamma|{\mathbb Z}$, which gives a counterexample if you take $\Gamma = G\times H$ for any $G$ and $H$ whose orders are not coprime.


