Finite reflection groups and linear preserver problems

Let G be one of the Coxeter groups A_n, B_n, D_n, or I₂(n), naturally acting on a Euclidean space V, and let ℒ(G) stand for the set of linear transformations ϕ of End V that satisfy ϕ(G) = G. It is easy to see that ℒ(G) contains all transformations of the form X ↦ PXQ, X ↦ PX*Q, where P, Q belong to the normalizer of G in the orthogonal group and PQ ∈ G. We show that in most cases these transformations exhaust ℒ(G); the only (rather unexpected) exception is the case G = B_n.