Given a fiber bundle $(E,B,p,F)$ with path connected base $B$ and fiber $F$, both closed smooth manifolds of finite dimensions. The second page $E_2^{p,q}$ of the Leray-Serre spectral sequence over $\mathbb{Z}_2$ is give by $H^p(B;\mathcal{H}^q(F;\mathbb{Z}_2))$, where $\mathcal{H}^q(F;\mathbb{Z}_2)$ is the local system (sheaf) of $\mathbb{Z}_2$-vector spaces on $B$ given by $\mathcal{H}^q(F;\mathbb{Z}_2)|_b=H^q(p^{-1}(b);\mathbb{Z}_2)$ for any $b\in B$. In the case this local system is trivial, namely, $\pi_1(B)$ acts trivially on it, we have $$E_2^{p,q}=H^p(B;\mathcal{H}^q(F;\mathbb{Z}_2))=H^p(B;\mathbb{Z}_2)\otimes H^q(F;\mathbb{Z}_2).$$

But if the local system is nontrivial, is there still any relation between two vector spaces $E_2^{p,q}$ and $H^p(B;\mathbb{Z}_2)\otimes H^q(F;\mathbb{Z}_2)$? More generally, is it true that the bi-graded $\mathbb{Z}_2$-algebra $E_2^{*,*}$ is isomorphic as bi-graded $\mathbb{Z}_2$-algebras to $\left(H^*(B;\mathbb{Z}_2)\otimes H^*(F;\mathbb{Z}_2)\right)/I$ for some ideal $I$?