Abstract For every finitely generated recursively presented group S we con- struct a finitely presented group Oi containing 9 such that 9 is (Frattini) embedded into Ji and the group % has solvable conjugacy problem if and only if 9 has solvable conjugacy problem. Moreover 9 and 3i have the same r.e. Turing degrees of the conjugacy problem. This solves a problem by D. Collins. Key words and phrases: Conjugacy problem, Higman embedding, fi- nitely presented group, recursively presented group, Turing degree 2000 Mathematics Subject Classification: 20E07, 20F06, 20F10
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