Finite abelian and abelian automorphism group implies cyclic

This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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Statement

Suppose $G$ is a Finite abelian group (?) that is also an Aut-abelian group (?), i.e., the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ is also a (finite) Abelian group (?). Then, $G$ is a Cyclic group (?) (and hence, a Finite cyclic group (?)).

Related facts

Similar facts

• Cyclic implies aut-abelian
• Aut-abelian implies class two
• Aut-abelian not implies abelian
• Aut-cyclic implies abelian
• Finite and aut-cyclic implies cyclic
• Classification of finite aut-cyclic groups

Opposite facts

• Abelian and aut-abelian not implies locally cyclic
• Aut-cyclic not implies cyclic

Proof

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