Let $G$ be a finite abelian group such that $G$ is of odd order or decomposition of the sylow 2-subgroups of $G$ there is at least two cyclic direct factors of maximal order. Then prove or disprove that for any subgroup $H$ of $G$ there exists an automorphism $\alpha$ of $G$ such that $H=\{g\in G| \alpha(g)=g\}$.
I think this is true.
Thank you