Certainly, such a gadget exists.  Let $\newcommand{\Pg}{B\mathrm{Pin}_n^{(m)}}\Pg$ be the homotopy fiber of the map $BO_n\to K(\mathbb{Z}/2, m+2)$ classifying the Stiefel-Whitney class.  If we realize the homotopy fiber as the path-space, then the map $\Pg\to BO_n$ is a Serre fibration.  Apply the Kan loop group construction to get a surjective map of simplicial groups which does what you want, I think.  (You don't ask for a *central* extension, and I don't think my construction gives one; the kernel of the map of simplicial groups is not going to be abelian, though it models $K(\mathbb{Z}/2,m)$.)

As for an explicit construction, there's quite a bit of work on giving explicit constructions of  $\mathrm{String}_n$, which sits in a central extension $K(\mathbb{Z},2)\to \mathrm{String}_n \to \mathrm{Spin}_n$.  I particularly like Chris Schommer-Pries's paper ["Central extensions of smooth $2$-groups and a finite dimensional string $2$-group"](http://arxiv.org/abs/0911.2483), which produces an extension of $2$-groups, from which one can extract an extension of topological or simplicial groups.