Reflection group action on a reflection hyperplane

Question

Let $G$ be a finite reflection group acting on a euclidean vector space $V$ of dimension $n$ and let $H$ be one of the reflection hyperplanes. Then $H$ is stabilized by some (reflection) subgroup of $G$. Which subgroup of $G$ acts on the $n-1$ dimensional subspace $H$? Intuitively, this should be the group $G'$ which is generated by all the roots which are orthogonal to the root corresponding to $H$. But which groups arise in such a way?

Answer 1

Any finite reflection group can arise in the way you describe. Take your desired $G'$ acting in a vector space $V'$, and let $V=V'\oplus \mathbb{R}$, and take $G=\langle G',w\rangle$ where $w$ acts as the identity on $V'$ and negation on $\mathbb{R}$.