Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left(  m_{s,t}\right)  _{\left( s,t\right)  \in S\times S}$.

Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) \in S^2$ satisfying $s \neq t$ and $m_{s,t} < \infty$.

For every $\left(  s,t\right)  \in \mathfrak{M}$, let
$c_{s,t}$ be an element of $\left\{  1,-1\right\}  $. Assume that:

* We have $c_{s,t}=c_{s^{\prime},t^{\prime}}$ for any two elements $\left(  s,t\right) $ and $\left(  s^{\prime},t^{\prime}\right)  $ of $\mathfrak{M}$ for which there exists some $q \in W$ satisfying $s^\prime = qsq^{-1}$ and $t^\prime = qtq^{-1}$.

* We have $c_{s,t}=c_{t,s}$ for each $\left(  s,t\right) \in \mathfrak{M}$.

Let $W^{\prime}$ be the group with the following generators and relations:

*Generators:* the elements $s\in S$ and an extra generator $q$.

*Relations:*
\begin{align*}
s^{2}  &  =1\ \ \ \ \ \ \ \ \ \ \text{for every }s\in S;\\
q^{2}  &  =1;\\
qs  &  =sq\ \ \ \ \ \ \ \ \ \ \text{for every }s\in S;\\
\left(  st\right)  ^{m_{s,t}}  &  =1\ \ \ \ \ \ \ \ \ \ \text{for every
}\left(  s,t\right)  \in \mathfrak{M} \text{ satisfying } c_{s,t}=1;\\
\left(  st\right)  ^{m_{s,t}}  &  =q\ \ \ \ \ \ \ \ \ \ \text{for every
}\left(  s,t\right)  \in \mathfrak{M} \text{ satisfying } c_{s,t}=-1.
\end{align*}

There is clearly a surjective group homomorphism $\pi:W^{\prime}\rightarrow W$
sending each $s\in S$ to $s$, and sending $q$ to $1$. There is also a group homomorphism $\iota:\mathbb{Z}/2\mathbb{Z} \rightarrow W^{\prime}$ which sends the generator of $\mathbb{Z}/2\mathbb{Z}$ to $q$.

> **Question.** Is $\iota$ injective? Equivalently, is the sequence
\begin{equation}
1\longrightarrow\mathbb{Z}/2\mathbb{Z}\overset{\iota}{\longrightarrow}W^{\prime
}\overset{\pi}{\longrightarrow}W \longrightarrow1
\end{equation}
exact? Equivalently, is $\left\vert \operatorname*{Ker}\pi\right\vert =2$ ?

**Background.** This would generalize at least one of the two "spin symmetric groups" to the situation of any Coxeter group. It would explain one of the results (Theorem 2.3 **(b)**) in [Alexander Postnikov and Darij Grinberg, *Proof of a conjecture of Bergeron, Ceballos and Labbé*](http://nyjm.albany.edu/j/2017/23-70.html), and prove a generalization of this result (Conjecture 6.1 **(b)**).

I have tried generalizing the [standard approach to constructing the spin symmetric groups by embedding them in the Hecke-Clifford algebra](https://arxiv.org/abs/1110.0263), but to no avail so far. Nor has the existing literature on central extensions of Coxeter groups been particularly helpful ([Howlett's *On the Schur multipliers of Coxeter groups*](http://onlinelibrary.wiley.com/doi/10.1112/jlms/s2-38.2.263/abstract) counts the extensions abstractly, but doesn't help understanding whether a given one exists; [Burichenko's *On extensions of Coxeter groups*](http://www.tandfonline.com/doi/abs/10.1080/00927879508825315?journalCode=lagb20) gives a criterion that I don't seem to properly understand, as it gives me wrong answers).