Permutation Groups and Ideals of Turing Degrees

We study degrees and degree spectra of groups \({\mathfrak{G}}_{\mathrm{I}}\) ${\mathfrak{G}}_{\mathrm{I}}$ defined on a set of permutations of the natural numbers ω whose degrees belong to a Turing ideal I. A necessary condition and a sufficient condition are stated which specify whether an arbitrary Turing degree belongs to the degree spectrum of a group \({\mathfrak{G}}_{\mathrm{I}}\) ${\mathfrak{G}}_{\mathrm{I}}$ . Nonprincipal ideals I for which the group \({\mathfrak{G}}_{\mathrm{I}}\) ${\mathfrak{G}}_{\mathrm{I}}$ has or does not have a degree are exemplified.