Let $G$ be a $p$-group of order $p^{n}\geq p^{7}$ and its automorphism group is elementary abelian $p$-group. Then, it is clear that $G$ is nilpotent of class $2$. However, the converse is not true in general (take for example the unitriangular group of $3\times 3$ upper triangular matrices over $\mathbb{F}_{p}$). In fact, Marta Morigi proves that there is a $p$-group of order $p^{7}$ and class $2$ and its automorphism group is elementary abelian $p$-subgroup of order $p^{12}$. Furthermore, I think the converse can be true for 2-groups $G$ with abelian direct factor but I don't know how to do this.

Are there any assumption to add to p-groups of class $2$ to get the converse for arbitrarily integer $n$?

Let $p$ be an odd prime and $G$ be a purely non-abelian p-group of class $2$ and order $p^{n}\geq p^{7}$. Does the automorphism group of $G$ is abelian?

Are there some other class of $p$-groups with elementary abelian automorphism $p$-groups?.

Any response or reference may be very helpful. Thank you in advance.

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