Local coefficient System and the action of fundamental group

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A local coefficient system $A\hookrightarrow E \to B$ is a fiber bundle $p:E\to B$ such that

  • The fiber is a discrete abelian group $A$
  • The structure group $G$ is a subset of Aut$(A)$

Is the action of $\pi_1(B)$ on $A$ necessarily a homomorphism?