Groups of permutations and ideals of Turing degrees

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