  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 it 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.
