Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$. Is $i^*\mathcal{I}$ still an injective sheaf on $Z$?  

The difficulty seems to lie in the fact that $Z$ is closed: I do not know the left-adjoint of $i^*$ (is it an extension by zero?).

Many thanks!