I am interested in a concept somehow dual to reflection subgroups. A *reflection quotient* of a Coxeter system $(W, S)$ shall be a surjective homomorphism $W \to W'$ to a Coxeter group $W'$ such that the images of $S$ are reflections. I did not find any literature on the topic, so it's possible that this notion has a name different from the one I just gave it. My question basically is the following:

Given a Coxeter system $(W, S)$, is there a way to find all its reflection quotients?

For $S_n$ (the Coxeter group with graph $A_{n-1}$), the answer is quite straightforward because $S_n$ has almost no normal subgroups: The only reflection quotients are $S_n$ and $S_2$. I do not know an answer for any other Coxeter groups, not even the finite ones. I am particularly interested in the Coxeter groups with graph $D_n$ (I even asked a specific question on it on MSE, but it did not receive any attention so far). The general problem however seems interesting enough for this forum, I hope.

One would hope that the problem has a graph-theoretic answer, similar to the one requested in the dual question to this one. I do not know how realistic this hope is.