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.