Lifting of elements of Weyl groups

Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N=Norm(T), and W=N/T is the Weyl group. If w∈W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.