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)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about group whose automorphism group is abelian|Get more facts about abelian group

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.

References

• A non-abelian group whose group of isomorphisms is abelian by G. A. Miller, Messenger Math., Volume 43, Page 124 - 125(Year 1913): ^{More info}
• Some non-abelian p-groups with abelian automorphism groups by David Jonah and Marc Konvisser, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 26, Page 131 - 133(Year 1975): ^{Official copyMore info}
• Some nonabelian 2-groups with abelian automorphism groups by Rebecca Roth Struik, Archiv der Mathematik, ISSN 1420-8938 (Online), ISSN 0003-889X (Print), Volume 39,Number 4, Page 299 - 302(Year 1982): ^{Official copyMore info}
• Some new non-abelian 2-groups with abelian automorphism groups by Ali-Reza Jamali, Journal of Group Theory, ISSN 14435883 (print), ISSN 14434446 (online), Volume 5,Number 1, Page 53 - 57(Year 2002): ^{Official copyMore info}