  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 $\mathcal{K}(E)$.
