On free topological groups

In the present article the existence and unicity of almost $(\mathcal U,\mathcal V)$-free group over given space, where $\mathcal U,\mathcal V$ are classes of topological groups is studied. If $\mathcal V$ is a quasivariety of compact topological groups and $\mathcal V\subseteq\mathcal U$, then these objects exist for any space. If $W$ is a quasivariety of compact groups, $\mathcal U=\mathcal V$ is the class of all pseudocompact subgroups of groups from $W$, then the almost $(\mathcal U,\mathcal V)$-free groups exist only for some special spaces.