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Consider the following natural question:

Given a finite group $H$, does there exists a finite group $G$ such that $\mathrm{Aut}(G)\cong H$?

In short, does any finite group occurs as the automorphism group (of some group)? The answer is NO. For example, $H=\mathbb{Z}_3$. Another counter-example is $H=Q_8$, the quaternion group of order $8$ (I leave their verification to interested reader). Note that $\mathbb{Z}_4$ occurs as automorphism group of $\mathbb{Z}_5$. This raises following two questions to me:

  1. Among finite abelian groups, which groups can occur as automorphism groups?

  2. Is there any other non-abelian finite group which can not occur as automorphism group?

Perhaps first question is easy to answer since structure of automorphism group of finite abelian groups is well known; I don't know its complete answer. For second question, I would be happy to see if there is an infinite family of counterexamples.

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There are several results addressing this question. MacHale proved $1983$ in his article Some Finite Groups Which Are Rarely Automorphism Groups

Theorem: There is no group $G$ such that $Aut(G)$ is abelian of order $p^5$, $p^6$ or $p^7$ for a prime $p$.

Theorem: There is no group $G$ such that $Aut(G)$ has order $p^4$, for odd prime $p$.

More recently, Ban an Yu showed:

Theorem: there is no group $G$ such that $Aut(G)$ is an abelian $p$-group of order $n<p^{12}$, where $p>2$ is a prime, but there is one with $|Aut(G)|=p^{12}$.

For the abelian case: There is no group $G$ such that $Aut(G)\cong C_n$ for $n>1$ odd, see here. A full classification of abelian groups occuring as automorphism groups of finite groups is still not known.

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