Free functor $\mathsf{sSet} \to \mathsf{sGrp}$ on homotopy groups The free functor $\mathsf{sSet} \to \mathsf{sAb}$ corresponds to the Hurewicz map on homotopy groups.
What about the free functor $\mathsf{sSet} \to \mathsf{sGrp}$? What effect does it have on homotopy groups? It has to abelianize $\pi_1$, since simplicial groups have all abelian homotopy groups, by an Eckmann-Hilton argument.
 A: If $K$ is a simplicial set, then the free simplicial group on $K$ is naturally weak equivalent to $\Omega\Sigma K_+$ (with the inclusion of $K$ into the free simplicial group corresponding to the natural inclusion $K\to K_+\to \Omega\Sigma K_+$).  More generally, if $K$ is a pointed simplicial set and you take the pointed free simplicial group on $K$ (where "pointed free" means that the basepoint of $K$ becomes the identity in the group), this is naturally weak equivalent to $\Omega\Sigma K$ (again, in a way compatible with the canonical maps from $K$).  See, for instance, Corollary V.6.17 in Goerss and Jardine's Simplicial Homotopy Theory.  The induced map on homotopy groups can thus be identified with the suspension map $\pi_n(K)\to\pi_{n+1}(\Sigma K)$.  Of course, unlike the case of the Hurewicz map, the codomain $\pi_{n+1}(\Sigma K)$ is typically not particularly easy to compute.
Morally, what's going on is that a group structure on a pointed space is equivalent (up to homotopy) to a loopspace structure (i.e., another space that it is the loopspace of).  So you should expect the free group on a space $K$ to be the initial loopspace that $K$ maps into.  A map $K\to \Omega X$ is the same as a map $\Sigma K\to X$, and the initial such $X$ is just $\Sigma K$, so the initial loopspace $K$ maps into is $\Omega\Sigma K$.
