TITLE: When is Aut(G) abelian
QUESTION [33 upvotes]: let $G$ be a group such that $\mathrm{Aut}(G)$ is abelian. is then $G$ abelian?
This is a sort of generalization of the well-known exercise, that $G$ is abelian when $\mathrm{Aut}(G)$ is cyclic, but I have no idea how to answer it in general. At least, the finitely generated abelian groups $G$ such that $\mathrm{Aut}(G)$ is abelian can be classified.

REPLY [7 votes]: A detailed survey on this problem for finite groups is now published 
 and is available at https://link.springer.com/chapter/10.1007/978-981-13-2047-7_7. A pre-published version is  available at https://arxiv.org/abs/1708.00615.
There have been some activities on this topic recently. No example of a non-special finite $p$-group having abelian automorphism group was know until quite recently. A class of such groups is constructed in "V. K. Jain, M. K. Yadav, On finite p-groups whose automorphisms are all central, Israel J. Math. 189 (2012), 225 - 236." This paper also contains a quick  survey of results on the topic and a big bibliography. Some more, different kind of examples are available at http://arxiv.org/pdf/1304.1974.pdf