Suppose that $X$ is a topological space, $\mathcal{F}$ is a sheaf (of abelian groups, rings, ideals, modules) on $X$ and $Y \subset X$. Do you know a natural way to get an induced sheaf on $Y$ from $\mathcal{F}$?
If $Y$ is open, the answer is straight forward. I will be happy with an answer assuming the following extra conditions: $Y$ is closed and $X$ is irreducible.