Finite abelian and abelian automorphism group implies cyclic

Statement
Suppose $$G$$ is a finite abelian group that is also a  group whose automorphism group is abelian, i.e., the automorphism group $$\operatorname{Aut}(G)$$ is also a (finite)  abelian group. Then, $$G$$ is a cyclic group (and hence, a  finite cyclic group).

Similar facts

 * Cyclic implies abelian automorphism group
 * Abelian automorphism group implies class two
 * Abelian automorphism group not implies abelian
 * Cyclic automorphism group implies abelian
 * Finite and cyclic automorphism group implies cyclic
 * Classification of finite groups with cyclic automorphism group

Opposite facts

 * Abelian and abelian automorphism group not implies locally cyclic
 * Cyclic automorphism group not implies cyclic