$\overline{MS}$-on-shell quark mass relation up to four loops in QCD and a general SU(N) gauge group

We compute the relation between heavy quark masses defined in the modified minimal subtraction and the on-shell schemes. Detailed results are presented for all coefficients of the SU(N$_c$) color factors. The reduction of the four-loop on-shell integrals is performed for a general QCD gauge parameter. Altogether there are about 380 master integrals. Some of them are computed analytically, others with high numerical precision using Mellin-Barnes representations, and the rest numerically with the help of FIESTA. We discuss in detail the precise numerical evaluation of the four-loop master integrals. Updated relations between various short-distance masses and the $\overline{MS}$ quark mass to next-to-next-to-next-to-leading order accuracy are provided for the charm, bottom and top quarks. We discuss the dependence on the renormalization and factorization scale