Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Question

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a compact $G_\delta$ with respect to $F$. Clearly $K$ is compact as a subset of $X$. Must it be a $G_\delta$ with respect to $X$?