On the automorphically stable distributions on Abelian groups

On a local compact Abelian group X, we consider G-automorphically stable distributions, where G is a subgroup of a group Aut(X). It is shown that if μ is G-automorphically stable, then 1) μ is either absolutely continuous, singular, or discrete with respect to the Haar measure of the group X; 2) if n is discrete, then μ 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 μ is a shift of the Haar distribution of a compact G-automorphically stable subgroup of the group X.