Finite-index extensions of essentially free countable Borel equivalence relations

Kechris asked whether essential freeness of countable Borel equivalence relations is preserved under extensions of finite index. Kaya proved that every such extension of a relation $E$ is Borel reducible to the equivalence relation $E^{\mathrm{f}\mathrm{i}\mathrm{n}}$ on nonempty finite subsets which records the $E$-classes met by the subset. We show that $E^{\mathrm{f}\mathrm{i}\mathrm{n}}$ is essentially free whenever $E$ is. This gives an affirmative answer to Kechris’s question.