Suppose that $G$ is a group acting on a fibre bundle $(F,E,B)$ by bundle automorphisms. In this case, the action automorphisms $E\to E$ give the integral homology $H_\ast(E;\mathbb{Z})$ the structure of a $G$-module. Also, the action automorphisms $F\to F$ and $B\to B$ give each module $H_p(B;H_q(F;\mathbb{Z}))$ the structure of a $G$-module. Can the Serre spectral sequence of the fibre bundle be made $G$-equivariant in the sense of being a spectral sequence of $G$-modules converging to $H_\ast(E;\mathbb{Z})$, and with second page $E_{p,q}=H_p(B;H_q(F))$ (considered as $G$-modules in the above sense)?

Thanks!