When the local system of coefficients are simple in the Leray-Serre spectral sequence

Question

Let $F\to E\to B$ a fibration and $\{E_{r}^{\ast,\ast},d_{r}\}$ the Leray-Serre Spectral sequence converging to $H^{\ast}(E;R),$ such that $$E_{2}^{p,q}=H^{p}(B;\mathcal{H}^{q}(F;R))$$ is the cohomology of $B$ with local coefficients in the cohomology of the fiber $F.$ We know that when the action of $\pi_{1}(B)$ induced by fibration on the cohomology of $F$ is trivial then the system of local coefficients is simple. My question: Can anything be said (assuming that $R=\mathbb{Z}_{2}$ and $\pi_{1}(B)=\mathbb{Z}_{2}$) when the action is trivial except for ONE element of the cohomology? To be more specific, there is only one element $c\in H^{1}(F;R)$ such that $\mu(g,c)=gc\neq c,$ for $g\in \mathbb{Z}_{2}=\langle g\rangle$, where $\mu:\pi_{1}(B)\times H^{\ast}(F)\to H^{\ast}(F)$ denotes the induced action.