The finitary power semigroup of a numerical semigroup S has only the identity automorphism unless S contains all integers from some k onward, in which case its automorphism group is generated by the identity and the involution X maps to max X minus X plus min X.
Hua, Geoometries of symmetric matrices over any field with characteristic other than two, Annals of Mathematics, 50 (1949), 8-31
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On automorphism groups of power semigroups over numerical semigroups or over numerical monoids
The finitary power semigroup of a numerical semigroup S has only the identity automorphism unless S contains all integers from some k onward, in which case its automorphism group is generated by the identity and the involution X maps to max X minus X plus min X.