Suppose that $A,B$ and $C$ are groups and that a fourth group $G$ acts on all three by automorphisms. If there is a short exact sequence $$ (*)\ \ \ \ \ \ 1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1 $$ compatible with the $G$-actions, is the (co)homological Hochschild-Serre spectral sequence for (*) (with $\mathbb{Q}$-coefficients, say) a spectral sequence of $G$-modules? I am sure this must be known, but I have not been able to find a reference and haven't quite been able to work it out myself.
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I assume this is a particular case of a Grothendieck spectral sequence. Identify the two functors $F$ and $G$ and compute (or at least identify is what I mean) their derived functors $R^pF$, $R^qG$ and $R^r(G \circ F)$. If all three constructions map $G$-modules to $G$-modules, you have your answer. Did you try doing that? – Patrick Da Silva Sep 26 '16 at 23:00