Finite Normal Semigroups

n

-element set X _n has only inner automorphisms is investigated. For a semigroup S of transformations of X _n, let G _s denote the group of all permutations of X _n that preserve S under conjugation. Fix a group G of permutations of X _n and let T be a semigroup of transformations of X _n maximal with respect to the property G _T = G. For certain groups G we give a complete description of all such T and prove that Aut T + Inn T ≅ G.