An Exercise in Allen Hatcher's book on Spectral Sequences

anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows:

For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ show that the associated action of $\pi_1K(C,1)=C$ on $H_*(K(A,1);G)$ is trivial if $A$, regarded as a subgroup of $B$, lies in the center of $B$.

Any help will be greatly appreciated.

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