General case I'm interested in: let $G = H_1 \times H_2$ where both $H_i$ are abelian groups. Let $M \in \operatorname{Mod}_{k[G]}$, then one has the spectral sequence $H^i(H_1, H^j(H_2, M))$ converging to $H^{i+j} (G, M)$. Does this spectral sequence degenerate at the second page?
More specific and, probably, more approachable case: Let $G = C_1 \times C_2$ where $C_1 = C_2 = \mathbb{Z}/p$ and $M \in \operatorname{Mod}_{\bar{\mathbb{F}_p}[G]}$ a module with the property that $H^i (C_1, M) = k$ for $i>0$ and as a $C_2$-module it is the trivial representation. Does the spectral sequence generate in this case? I believe it follows from the works of Cartan/Serre/Leary that there are no differentials on pages $2, 3,$ and $4$ but I do not see a general argument to show that it degenerates.
I would be grateful for a reference discussing degeneration of the spectral sequence for non-constant coefficients in general.