Finite index subgroups of conjugacy separable groups

We construct an example of conjugacy separable group possessing a subgroup of finite index which is not conjugacy separable. We give also a sufficient condition for a conjugacy separable group to preserve this property when passing to subgroups of finite index. We establish also conjugacy separability of finitely presented residually free groups using impressive results of Bridson and Wilton [Subgroup separability in residually free groups].