Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.

I am looking for a counterexample for the following statement:

$E \cong \widetilde{B} \times_G F'$, where $G$ is a finite quotient of $\pi_1(B)$ acting on a compact smooth manifold $F'$ and $\widetilde{B}$ is the cover of $B$ associated to that quotient. "$\cong$" stands for a) fiberwise diffeomorphic, b) fiberwise homotopy equivalent.

**Related**

This is one aspect of this question Compute cohomology of flat fiber bundles - does this always work?, more precisely formulated.