Permutation groups with the same movement

Let G be a permutation group on a set Omega with no fixed point in Omega. If for each subset Gamma of Omega the size |Gamma(g) \ Gamma| is bounded, for g is an element of G, we define the movement of g as the max |Gamma(g) \ Gamma| over all subsets Gamma of Omega, and the movement of G is defined as the maximum of move(g) over all non-identity elements of g is an element of G. In this paper we classify all permutation groups with maximum possible degree in which every non-identity element has the same movement.