Separability properties in groups can be seen as an algebraic analogue of classical decision problems in finitely presented groups, as an argument by Mal'cev shows that if a subset S of a finitely presented group is recursively enumerable, separable and its image under a surjective homomorphism to a finite group can be computed, then we can decide membership on S.
We will introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy and orbit decidability problems and show how they relate when finite and cyclic extensions of groups are taken. As it happens with the algorithmic versions of the problems, generalized twisted conjugacy separability of a group G with respect to virtually inner automorphisms is equivalent to generalized conjugacy separability in finite extensions of G and, similarly, generalized conjugacy separability in cyclic extensions of G implies generalized twisted conjugacy and orbit separability in G.

Applications include results in free, virtually abelian, virtually polycyclic groups and a proof that virtually free times free groups are conjugacy separable.