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, 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, 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 counts the extensions abstractly, but doesn't help understanding whether a given one exists; Burichenko's On extensions of Coxeter groups gives a criterion that I don't seem to properly understand, as it gives me wrong answers).
