$G_{\delta }$ sets in $\sigma$-ideals generated by compact sets

Given a compact Polish space $E$ and the hyperspace of its compact subsets $\mathcal{K}(E)$, we consider the class of $G_{\delta }$ $\sigma$-ideals of compact subsets of $E$ that can be represented via a compact subset of $\mathcal{K}(E)$. If we extend such an ideal $I$ by considering $G_{\delta }$ (or analytic) sets that are covered by countable unions of sets in $I$, we show that the extended collection can still be represented via some compact subset of $\mathcal{K}(E)$.