# Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on $H_\ast(F)$ is trivial. My question is the following:

Except when $\pi_1(B)$ is zero, what simple conditions are there for ensuring that the action is trivial? Does say, nilpotency of $\pi_1(B)$ suffice?