The automorphism group of reduced power monoids of finite abelian groups

Let $H$ be an additively written monoid and let ${\mathcal{P}}_0(H)$ denote the reduced power monoid of $H$, that is, the monoid consisting of all subsets of $H$ containing $0$ with set addition as operation. Following work of Tringali, Wen and Yan, we give a full description of the automorphism group of ${\mathcal{P}}_0(G)$, where $G$ is a finite abelian group. More precisely, we show that $\text{Aut}({\mathcal{P}}_0(G))$ and $\text{Aut}(G)$ are isomorphic in a canonic way, except in the special case when $G$ is isomorphic to the Klein four-group.