On the automorphisms of the power semigroups of a numerical semigroup

If $H$ is a numerical semigroup (that is, a cofinite subset of the non-negative integers closed under addition), then the non-empty subsets of $H$ form a semigroup $\mathcal{P}(H)$ under the sumset operation induced by addition in $H$. Moreover, if $0\in H$, then $\mathcal{P}(H)$ is a monoid with identity element $\{0\}$, and the family ${\mathcal{P}}_0(H)$ of all subsets of $H$ containing $0$ is a submonoid of $\mathcal{P}(H)$. We show that the automorphism group of $\mathcal{P}(H)$ is trivial, and the same holds for ${\mathcal{P}}_0(H)$ when $0\in H$. The proofs blend ideas from combinatorics and semigroup theory.