Countable Borel equivalence relations are ubiquitous objects in descriptive set theory, encoding orbit structures of countable group actions on standard Borel spaces. An equivalence relation is essentially free if it is Borel reducible to one generated by a free group action. When one builds a finite-index extension of such a relation, meaning a new equivalence relation that refines the original by splitting some classes into finitely many subclasses, it is not known in general whether essential freeness is preserved. The question asks for a complete algebraic or combinatorial characterization of when finite-index extensions inherit this property and when they do not.