We establish key connections between Green’s \({\mathscr {J}}\) - and \({\mathscr {L}}\) -relations on a finite semigroup and the subduction relation defined on the image sets of an action of the same semigroup when it acts faithfully on a finite set. The construction of the skeleton order, the partial order on equivalence classes of the subduction relation, is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the Green’s \(\le _{\mathscr {L}}\) -order and \(\le _{\mathscr {J}}\) -order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the \({\mathscr {J}}\) -class order and the skeleton order is one of isomorphism. Finally, we characterize the relationship between natural subsystems of a transformation semigroup, permutator groups and the \({\mathscr {H}}\) -relation.