Families of permutations and ideals of Turing degrees

Families ${\mathcal P}_{\mathrm I}$ consisting of permutations of the natural numbers $\omega$ whose degrees belong to an ideal $\mathrm I$ of Turing degrees, as well as their jumps ${\mathcal P}'_{\mathrm I}$, are studied. For any countable Turing ideal $\mathrm I$, the degree spectra of families ${\mathcal P}_{\mathrm I}$ and their jumps ${\mathcal P}'_{\mathrm I}$ are described. For some ideals $\mathrm I$ generated by c.e. degrees, the spectra of families ${\mathcal P}_{\mathrm I}$ are defined.