Semigroups of transformations whose restrictions belong to a permutation group

Let [Formula: see text] be a set and [Formula: see text] the full transformation semigroup on [Formula: see text]. For each nonempty subset [Formula: see text] of [Formula: see text], we consider the semigroup [Formula: see text] of all transformations on [Formula: see text] whose restrictions to [Formula: see text] belong to a permutation group [Formula: see text] on [Formula: see text], namely, [Formula: see text] In this paper, we characterize Green’s relations and ideals of [Formula: see text]. We show that the minimal ideal of [Formula: see text] is a left group and is a union of permutation groups. Moreover, a generating set of [Formula: see text] is determined. Finally, the set of all minimal idempotents of [Formula: see text] is described and then we prove isomorphism theorems for this semigroup.