Let $C$ be a Serre class which satisfies the additional axioms about $\otimes, \mathrm{Tor}, K(A,1)$'s.
It is then easy to check that if $F\to X\to B$ is a Serre fibration with $\pi_1(B)$ acting trivially on the homology of $F$, and the (positive) homologies of $B,F$ are in $C$, then the same is true of $X$.
(For that we don't even need $K(A,1)$'s)
Indeed it follows easily from the Serre spectral sequence and the universal coefficient theorem applied on $E^2$.
I'm wondering if this holds when $\pi_1(B)$ acts nontrivially.
Is it true that under the hypotheses (minus the hypothesis on the action) $H_p(B, H_q(F)) \in C$ ? (Here it's therefore homology with local coefficients)
If not, does $H_p(X) \in C$ still hold for some other reason ?
(I'm only interested in homology in strictly positive degrees)
EDIT : more specifically, here are all the axioms for $C$:
1) if $0\to M\to N\to L\to 0$ is a short exact sequence, $N\in C \iff M,L \in C$
2) If $A,B\in C, A\otimes B, \mathrm{Tor}(A,B) \in C$
3) If $A\in C$, for all $k>0, H_k(K(A,1))\in C$
Note that if there is a counterexample where $C$ doesn't satisfy 3), I'm also interested
2nd Edit : William answered the question I asked, but I'm wondering if anyone has an example with a connected fiber, so I'll leave this as unaccepted for a couple of days to see if one can find an example with connected fiber.
Best Answer
What about the bundle $\mathbb{Z}/2 \to S^n \to \mathbb{R}P^n$ where $n>0$ is even? Then for all $k>0$ the groups $H_k(\mathbb{Z}/2)$ and $H_k(\mathbb{R}P^n)$ are all in the Serre class of finite groups, but $H_n(S^n)\cong \mathbb{Z}$.