In the paper "Continuous isomorphisms onto separable groups", Applied General Topology, (13) 2012, 135--150, L. Morales Lopez proved Theorem: Let $G$ be an Abelian group with $|G| \leq 2^{2^{\aleph_0}}$. Then $G$ admits a separable, precompact, Hausdorff group topology. It is not true for general non-abelian groups by Shelah's results.

Is it true that any solvable group admits a separable Hausdorff group topology? Are there any published results about sufficient conditions under which a group admits a separable Hausdorff group topology?