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 $\mathbb{Z}^k$.



Kunneth formula for semidirect product
Just to collect the references I wrote in comments, and more. They each give (in)finite (non)abelian counterexamples, as well as general explanations for failure of collapse of the LHS spectral sequence with semi-direct products: 


1. *Charlap, L. S.; Vasquez, A. T.*, [**The cohomology of group extensions**](http://dx.doi.org/10.1090/S0002-9904-1963-11046-0), Bull. Am. Math. Soc. 69, 815-817 (1963). [ZBL0122.02803](https://zbmath.org/?q=an:0122.02803).
2. *Benson, D. J.; Feshbach, M.*, [**On the cohomology of split extensions**](http://dx.doi.org/10.2307/2160262), Proc. Am. Math. Soc. 121, No. 3, 687-690 (1994). [ZBL0819.20058](https://zbmath.org/?q=an:0819.20058).
3. *Totaro, Burt*, [**Cohomology of semidirect product groups**](http://dx.doi.org/10.1006/jabr.1996.0181), J. Algebra 182, No. 2, 469-475 (1996). [ZBL0862.20038](https://zbmath.org/?q=an:0862.20038).
4. *Siegel, Stephen F.*, [**On the cohomology of split extensions of finite groups**](http://dx.doi.org/10.1090/S0002-9947-97-01747-9), Trans. Am. Math. Soc. 349, No. 4, 1587-1609 (1997). [ZBL0951.20039](https://zbmath.org/?q=an:0951.20039).


See also references within.


