  Let ${\cal C}$ be a nonempty class of finite groups closed under taking
subgroups, homomorphic images and extensions. A subgroup $H$ of an abstract
residually ${\cal C}$ group $R$ is said to be conjugacy ${\cal
C}$-distinguished if whenever $y\in R$, then $y$ has a conjugate in $H$ if and
only if the same holds for the images of $y$ and $H$ in every quotient group
$R/N\in {\cal C}$ of $R$. We prove that in a group having a normal free
subgroup $\Phi$ such that $R/\Phi$ is in ${\cal C}$, every finitely generated
subgroup is conjugacy ${\cal C}$-distinguished. We also prove that finitely
generated subgroups of limit groups, of Lyndon groups and certain one-relator
groups are conjugacy distinguished (${\cal C}$ here is the class of all finite
groups).
