  In this paper we consider the {\em conjugacy stability} property of subgroups
and provide effective procedures to solve the problem in several classes of
groups. In particular, we start with free groups, that is, we give an effective
procedure to find out if a finitely generated subgroup of a free group is
conjugacy stable. Then we further generalize this result to quasi-convex
subgroups of torsion-free hyperbolic groups and finitely generated subgroups of
limit groups.
