Twisted conjugacy and separability

A group $G$ is twisted conjugacy separable if for every automorphism $\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for any group $H$ and any two homomorphisms $\varphi,\psi$ from $H$ to $G$, distinct $(\varphi,\psi)$-twisted conjugacy classes can be separated in a finite quotient. We study how these properties behave with respect to taking subgroups, quotients and finite extensions, and compare them to other notions of separability in groups. Finally, we show that for polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy separable is equivalent to all quotients being residually finite.