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Suppose we have a (topological) fiber bundle $p:E\to B$ with fiber $F$ and structure group $G$. Since $G$ acts on $F$ by homeomorphisms, it induces an action on the (integral) homology $H_*(F)$, i.e., there is a group homomorphism $\phi:G\to\operatorname{Aut}(H_*(F))$.

The bundle $E$ also comes with a natural action of the fundamental group $\pi_1(B)$ on the (integral) homology of the fiber, giving the monodromy representation $\psi:\pi_1B\to\operatorname{Aut}(H_*(F))$.

Question. What is the relationship between the images of these two representations in $\operatorname{Aut}(H_*(F))$? In particular, is it necessarily true that the image of the monodrompy representation $\psi$ lies in the image of $\phi$ (i.e., $\pi_1(B)$ "acts by" elements of the structure group $G$)?
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