What are the known "families of examples" of non-abelian p-groups with abelian groups of automorphisms?

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Vitaly Roman'kov

There is a lot of examples of various types of finite p-graups having abelian automorphism group. In 1913, the first example a non-abelian group of order 64 such that Aut(G) is an elementary abelian group of order 128 has been constructed by G.A. Miller. A comprehensive survey about such groups can be found in: Vivek K. Jain, Pradeep K. Rai, and Manoj K. Yadav's preprint in arXiv: 1304.1974v1 [math. GR] 7 Apr 2013.

Rahul Dattatraya

@Vitaly: Thanks for the important suggestions on the question.