Let $\pi: X \to Y$ be a morphism of sites (I need this for the étale topology) and $Z \hookrightarrow Y$ be a closed subscheme.

Since an injective sheaf is flabby (in the sense that $H^i(U,F) = 0$, $i > 0$ for all $U$ of a site), and the pushforward of a flabby sheaf is flabby, it follows from the long exact localisation sequence that $H^i_Z(Y,\pi_*F) = 0$ for $i > 1$.

Now my question is: Does this also hold for $i = 1$? (Equivalently, is $(\pi_*F)(Y) \to (\pi_*F)(U)$ surjective for all $U$ and $F$ injective?

For the Zariski topology, this can be proved as follows: Let $U \hookrightarrow X$ be open, $F$ an injective sheaf on $X$. Then one has an exact sequence $0 \to j_!\mathcal{O}_U \to \mathcal{O}_X$ and $F(X) = \mathrm{Hom}(\mathcal{O}_X, F) \to \mathrm{Hom}(j_!\mathcal{O}_U, F) = \mathrm{Hom}(\mathcal{O}_U, F|_U) = F(U)$. Here, the middle arrow is surjective.)