If is an orbifold presented by an orbifold groupoid and is a finitely generated group, then induces a decomposition of into the -sectors of , a finite disjoint union of orbifolds including as well as lower-dimensional orbifolds related to the singular strata of . Specifically, the -sector decomposition is given by the space of groupoid homomorphisms from into , which admits a natural -action. This construction, which generalizes the inertia orbifold and orbifold of multi-sectors, was introduced by Tamanoi for global quotient orbifolds, i.e. orbifolds given by the quotient of a manifold by a finite group. Applying an orbifold invariant to the space of -sectors of yields the -extension of , a new invariant for orbifolds associated to each .

We will discuss the -extensions of the Euler-Satake characteristic. In particular, we will consider these invariants for - and -dimensional orbifolds as well as for wreath symmetric products of orbifolds, the orbifold analogue of the symmetric product of a manifold.