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?