TITLE: Kunneth formula for semidirect product
QUESTION [5 upvotes]: 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 $\mathbb{Z}^k$.

REPLY [7 votes]: 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:   

Charlap, L. S.; Vasquez, A. T., The cohomology of group extensions, Bull. Am. Math. Soc. 69, 815-817 (1963). ZBL0122.02803.
Benson, D. J.; Feshbach, M., On the cohomology of split extensions, Proc. Am. Math. Soc. 121, No. 3, 687-690 (1994). ZBL0819.20058. 
Totaro, Burt, Cohomology of semidirect product groups, J. Algebra 182, No. 2, 469-475 (1996). ZBL0862.20038.
Siegel, Stephen F., On the cohomology of split extensions of finite groups, Trans. Am. Math. Soc. 349, No. 4, 1587-1609 (1997). ZBL0951.20039.

See also references within.