The Kan-Thurston theorem obsesses me recently, which is proved by Daniel Kan and William Thurston, Every connected space has the homology of a K(π,1), Topology Vol. 15. pp. 253–258, 1976. Later, there are several similar results like some done by C. R. F. Maunder (A Short Proof of a Theorem of Kan and Thurston), G. Baumslag, E. Dyer, A. Heller (The topology of discrete groups), McDuff etc. See here for a related discussion.
I think (for simplicial sets), the Kan-Thurston result can be made functorial (although they only said functoriality for reduced simplicial sets). I wonder if there is a nice description （e.g. somewhat explicit, functorial) about the groups $G_X$ and $P_X$ in that paper (which I think are not unique—of course, their quotient is unique, being the fundamental group of the given space or simplicial set—but I guess there should be nice models).
$\mathbf{Edit}:$ Maybe for now I mainly want to know if the Kan-Thurston construction preserves homotopy: if $g_0, g_1: X\rightrightarrows Y$ are homotopic, then are $Tg_0, Tg_1: TX\rightrightarrows TY$ also homotopic? Recall that $TX$ is a $K(G_X, 1)$ in their construction.
