Suppose that G is a group acting on a fibre bundle (F,E,B) by bundle automorphisms. In this case, the action automorphisms E-->E give the integral homology H_{}(E;Z) the structure of a G-module. Also, the action automorphisms F-->F and B-->B give each module H_p(B;H_q(F;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_{}(E;Z), and with second page E_{p,q}=H_p(B;H_q(F)) (considered as G-modules in the above sense)?

Thanks!

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!