Group closures of one-to-one transformations

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S' under conjugation. Fix a permutation group H on X and a transformation f of X, and let (f : H) = ({hf h(-1) : h epsilon H}) be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G((f:H)) = H. We also show that if S is a semigroup of one-to-one transformations of X and G(S) contains the alternating group on X then Aut(S) = Inn(S) congruent to G(S).