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

Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. If for each subset Γ of Ω the size |ΓᵍΓ| is bounded, for g∈ G, we define the movement of g as the max|ΓᵍΓ| over all subsets Γ of Ω, and the movement of G is defined as the maximum of move$(g)$ over all non-identity elements of g∈ 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 $m-2$.