Here's something that might be considered an answer.

In Section 11 of Baumslag, Dyer, Heller, "The topology of discrete groups", the following theorem is proved, basically building on the ideas of the Kan-Thurston theorem.  

A *perfect* homomorphism of groups $G\to H$ is a surjective group homomorphism such that the kernel is a perfect group.  Let $\mathcal{C}$ be the category whose objects are perfect homomorphisms, and whose maps are commutative squares. 

The theorem is:

> There exists a class $\mathcal{F}$ of morphisms in $\mathcal{C}$ such that the category of fractions $\mathcal{C}[\mathcal{F}^{-1}]$ is equivalent to the homotopy category of pointed connected CW complexes.  

The functor from $\mathcal{C}$ to spaces is defined by the Quillen plus construction to $BG$ with respect to $P=\mathrm{Ker}(G\to H)$, and the class $\mathcal{F}$ is just the maps that become equivalences after applying the plus construction; the class $\mathcal{F}$ can also be characterized as the maps $\phi\colon (G'\to H')\to (G\to H)$ such that $H'\to H$ is an isomorphism and the induced maps $H_*(G',\phi^*M)\to H_*(G,M)$ in group homology are isos for every $G$-module $M$.