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Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.

It is proved that an arbitrary descending HNN-extension of a finitely generated Abelian group is conjugacy separable.

A sufficient condition for the residual p-finiteness (approximability by the class \(\mathcal{F}_p\) of finite p-groups) of a free product G = (A * B; H) of groups A and B with a normal amalgamated subgroup H is obtained. This condition is used to prove that if A and B are extensions of residually \(\mathcal{N}\)-groups by \(\mathcal{F}_p\)-groups, where \(\mathcal{N}\) stands for the class of finitely generated torsion-free nilpotent groups, and if H is a normal p′-isolated polycyclic subgroup, then the group G is residually p-finite (i.e., residually \(\mathcal{F}_p\)-group), provided the quotient group G/HpH′ is residually p-finite.