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