Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A Aut(F ) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z2 Fm, F2 Fm, Fn Z, and Zn A Fm with virtually solvable action group A GLn(Z). Also, w...