This seems to work (even though it looks a little bit too simple). To be specific I am using May's notation with $\newcommand{\hw}{\overline{W}}\hw$ for the classifying space of a simplicial group and $\newcommand{\kl}{\mathbf{G}}\kl$ for the Kan loop group. Start as Charles does with the the map $\hw\mathbf{O}_n\rightarrow \hw\hw\mathrm{K}(\mathbb{Z}/2,m)$. Applying the Kan loop construction to this map we get a map of simplicial groups $\kl\hw\mathbf{O}_n\rightarrow \kl\hw\hw\mathrm{K}(\mathbb{Z}/2,m)$ and as $G$ and $\hw$ are adjoint we have a map of simplicial groups $\kl\hw\hw\mathrm{K}(\mathbb{Z}/2,m)\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$ and composing we get a map of simplicial groups $\kl\hw\mathbf{O}_n\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$. Now, there is an exact sequence of abelian simplicial groups $0\rightarrow\mathrm{K}(\mathbb{Z}/2,m)\rightarrow H\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)\rightarrow0$ with $H$ contractible. Pulling back this along $\kl\hw\mathbf{O}_n\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$ gives a central extension $1\rightarrow\mathrm{K}(\mathbb{Z}/2,m)\rightarrow H'\rightarrow\kl\hw\mathbf{O}_n\rightarrow1$ which has the right homotopy type.

Note that this goes more or less backwards in the following observations: If $1\rightarrow Z\rightarrow G\rightarrow H\rightarrow1$ is a central extension, then as multiplication $Z\times G\rightarrow G$ is a group homomorphism we get a map $\hw Z\times\hw G\rightarrow\hw G$ which is an action of the simplicial group $\hw Z$ on $\hw G$ making $\hw G\rightarrow\hw H$ a $\hw Z$-torsor giving a classifying map $\hw H\rightarrow\hw\hw Z$.

This seems to work for getting a central extension (even though it looks a little bit too simple). To be specific I am using May's notation with $\newcommand{\hw}{\overline{W}}\hw$ for the classifying space of a simplicial group and $\newcommand{\kl}{\mathbf{G}}\kl$ for the Kan loop group. Start as Charles does with the the map $\hw\mathbf{O}_n\rightarrow \hw\hw\mathrm{K}(\mathbb{Z}/2,m)$. Applying the Kan loop construction to this map we get a map of simplicial groups $\kl\hw\mathbf{O}_n\rightarrow \kl\hw\hw\mathrm{K}(\mathbb{Z}/2,m)$ and as $G$ and $\hw$ are adjoint we have a map of simplicial groups $\kl\hw\hw\mathrm{K}(\mathbb{Z}/2,m)\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$ and composing we get a map of simplicial groups $\kl\hw\mathbf{O}_n\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$. Now, there is an exact sequence of abelian simplicial groups $0\rightarrow\mathrm{K}(\mathbb{Z}/2,m)\rightarrow H\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)\rightarrow0$ with $H$ contractible. Pulling back this along $\kl\hw\mathbf{O}_n\rightarrow\hw\mathrm{K}(\mathbb{Z}/2,m)$ gives a central extension $1\rightarrow\mathrm{K}(\mathbb{Z}/2,m)\rightarrow H'\rightarrow\kl\hw\mathbf{O}_n\rightarrow1$ which has the right homotopy type.

Note that this goes more or less backwards in the following observations: If $1\rightarrow Z\rightarrow G\rightarrow H\rightarrow1$ is a central extension, then as multiplication $Z\times G\rightarrow G$ is a group homomorphism we get a map $\hw Z\times\hw G\rightarrow\hw G$ which is an action of the simplicial group $\hw Z$ on $\hw G$ making $\hw G\rightarrow\hw H$ a $\hw Z$-torsor giving a classifying map $\hw H\rightarrow\hw\hw Z$.