Consider a surjective map $f\colon X\to Y$ of smooth projective varieties. It is well known (see e.g. Voisin's Hodge theory I, Lemma 7.28) that the map $H^i(Y,\mathbb Q)\to H^i(X,\mathbb Q)$ is injective.

Question: Can we replace $\mathbb Q_Y$ with some other sheaf $L$ in this statement? Namely, is it true that the map $$ H^i(Y,L)\to H^i(X,f^*L) $$ is surjective for, say, $L$ local system or of geometric origin, or constructible?

Remark: $X$ and $Y$ are over $\mathbb C$ and cohomology is Betti cohomology.