$\Gamma$-extensions of the spectrum of an orbifold

We introduce the $\Gamma$-extension of the spectrum of the Laplacian of a Riemannian orbifold, where $\Gamma$ is a finitely generated discrete group. This extension, called the $\Gamma$-spectrum, is the union of the Laplace spectra of the $\Gamma$-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the $\Gamma$-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that, in many cases, isospectral orbifolds need not be $\Gamma$-isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are $\Gamma$-isospectral for any choice of $\Gamma$.