On the automorphically stable distributions on Abelian groups

On a local compact Abelian group $X$, we consider {$G$-auto}-morphically stable distributions, where $G$ is a subgroup of a group $Aut(X)$. It is shown that if $\mu$ is $G$-automorphically stable, then 1) $\mu$ is either absolutely continuous, singular, or discrete with respect to the Haar measure of the group $X$; 2) if $\mu$ is discrete, then $\mu$ is a shift of the Haar distribution of a finite $G$-characteristic subgroup of the group $X$; 3) if $G$ consists of elements of finite order, then $\mu$ is a shift of the Haar distribution of a compact $G$-automorphically stable subgroup of the group $X$.