Transitive permutation groups with elements of movement m or m- 2

Let G be a permutation group on a set Q with no fixed points in Q and let m be a positive integer. If for each subset F of Q the size Fg g \ F is bounded, for g E G, we define the movement of g as the max Fg g \ F over all subsets F of Q, , and the movement of G is defined as the maximum of move(g) g ) over all non-identity elements of g E G. In this paper we classify all transitive permutation groups with bounded movement equal to m that are not a 2-group, but in which every non-identity element has movement m or m - 2.