  Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free
topological group and the free Abelian topological group over $X$ in the sense
of Markov. In this paper, we provide some topological properties of $X$
whenever one of $F(X)$, $A(X)$, some finite level of $F(X)$ and some finite
level of $A(X)$ is $q$-space (in particular, locally $\omega$-bounded spaces
and $r$-spaces), which give some partial answers to a problem posed in [11].
