Conjugacy separability and outer automorphism groups of certain HNN extensions

In this note we first establish a criterion for the conjugacy separability of HNN extensions of polycyclic-by-finite groups with central associated subgroups. Using this result, we prove the equivalence of residually finiteness and conjugacy separability in these HNN extensions. Then we prove that the outer automorphism groups of these HNN extensions are residually finite if the HNN extensions themselves are residually finite.