pushforward of injective sheaf acyclic for cohomology with supports

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.)

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It seems to the me that the étale case should reduced to the Zariski case, because the pushforward from the étale site to the Zariski site is also flabby. Am I missing something? –  Angelo Apr 20 '13 at 9:35
A variation on Angelo's question: where does the argument in the last paragraph use the Zariski topology? All you need is that $\pi_* F(Y) \to \pi_* F(Y - Z)$ is surjective, which is the same as asking that $F(X) \to F(X - f^{-1} Z)$ be surjective, which is what the last paragraph shows. –  anon Apr 20 '13 at 17:12
Thanks to both of you. –  Timo Keller Apr 22 '13 at 14:59