I am learning the Serre spectral sequence and I am intrigued about the degeneracy of such at the $E_2$-page. Assuming field coefficients in cohomology for simplicity.
In fact, for a Serre fibration $F \rightarrow E \rightarrow B$ it is known that the Serre spectral sequence degenerates at the $E_2$-page and the coefficient system is simple if and only if the edge homomorphism $H^*(E) \rightarrow H^*(F)$ is surjective. In this case, $H^*(E) \cong H^*(B) \otimes H^*(F)$ and thus $H^*(E)$ is a free module over $H^*(B)$.
I am wondering whether some sort of converse holds.
The projection map $E \rightarrow B$ induces on $H^*(E)$ a structure of $H^*(B)$-module and suppose further that it is a free module. Does it follow that the Serre spectral sequence degenerates at $E_2$?
I am aware of the Eilenberg-Moore spectral sequence which provides an answer when $B$ is simply connected. How about the situations where the coefficient system is not simple?