Extending a sheaf by zero outside a locally closed subspace

Question

Suppose $X$ is a locally closed subset of $Y$($X$can be expressed as an intersection of a closed set $F$and an open set $O$), $\mathcal{F}$ is a sheaf on Y, $\mathcal{F}|_X$ is its restriction on X, can we prove that there exists a unique sheaf $\mathcal{G}$ on $Y$ such that $\mathcal{F}|_{Y-X}$ is zero and $\mathcal{G}|_X=\mathcal{F}|_X$?