Normalizers of Sylow subgroups in finite reflection groups

Let W be a finite reflection group, either real or complex, and Sl a Sylow l-subgroup of W. We prove the existence of a semidirect product decomposition of NW(Sl) in terms of the unique parabolic subgroup of W minimally containing Sl and known decompositions of normalizers of parabolic subgroups. In the real setting, the description follows from the existence of Sylow l-subgroups stable under the Coxeter diagram automorphisms of finite reflection groups with no proper parabolic subgroup containing a Sylow l-subgroup.