Consider a short (not necessarily split) exact sequence of groups

$1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$

and suppose we wish to find the cohomology of $G$ with coefficients in a ring $R$. Then, it is known that there is a first quadrant cohomological spectral sequence **of algebras** converging **as an algebra**:

$E_2^{p,q} = H^p(Q;H^q(N;R)) \implies H^{p+q}(G;R)$.

Suppose $R$ is replaced by a $Q$-module $M$. (A specific example of interest to me is when $Q = \mathbb Z_2$ and $M = \mathbb Z$ with the inversion action of $Q$.) Then, can we still define a spectral sequence of algebras with an $E_2$-page as given above, so that it converges to $H^{p+q}(G,M)$ as an algebra?

(Knowing the answer for the specific example I mentioned is sufficient.)