Abelian automorphism group not implies abelian

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose automorphism group is abelian) need not satisfy the second group property (i.e., abelian group)
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Statement

There exist non-abelian groups (in fact, non-abelian finite $p$-groups for every prime $p$) that are groups whose automorphism group is abelian: the automorphism group is an abelian group.