I wonder if the following Kunneth formula for semidirect product is valid $$ H^n(N\rtimes_\phi G;\mathbb{Z}) = \sum_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$ where $H^*$ is the group cohomology and $G$ has a proper action on $H^j(N;\mathbb{Z})$ as induced by $\phi$. (For direct product, $G$ has no action on $H^j(N;\mathbb{Z})$ and the above reduces to the standard Kunneth formula.)

https://arxiv.org/abs/math/0406130 only showed above when $N$ has a form $\mathbb{Z}^k$.

"Cohomology of Semidirect Product Groups", and a disproof of a conjecture of Adem (arxiv.org/pdf/1105.4772.pdf) which I think is related to the arXiv paper you linked. $\endgroup$ – Chris Gerig Jul 27 '18 at 1:25