Separable quotients of free topological groups

We study the following problem: For which Tychonoò spaces X do the free topological group F(X) and the free abelian topological group A(X) admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? he existence of the required quotient homomorphisms is established for several important classes of spaces X, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of σ-compact spaces, the class of connected locally connected spaces, and some others. We also show that there exists an infinite separable precompact topological abelian group G such that every quotient of G is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.