Spectral sequence for equivariant group extension

Question

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.