Given a fibration $F \to E \to B$, under what circumstances does the inclusion of the homotopy fiber into $E$, $F \to E$, induce injections on homology? The specific case I'm dealing with involves the fibration $F_n \to X_n \to X_{n-1}$ from a Postnikov tower of a space $X$ satisfying the condition that $X$ is path-connected and abelian, so the action of $\pi_1(X)$ on $\pi_n(X)$ is trivial for all $n>1$.
Under these conditions, I know that $F_n$ is a $K(\pi_nX,n)$, and the fact that $X$ is abelian means that $X$ has a CW approximation whose Postnikov tower consists of principal fibrations $X_n \to X_{n-1}$, but don't see how this helps me prove that the map on homology induced by inclusion, $H_n(F_n) \to H_n(X_n)$, is injective, which is what I'm trying to prove.
Of course, I realize that this induced map on homology map not be injective, in general, in which case I'd be grateful for a counterexample. Either way, thank you for your time.