Every finite group occurs as the automorphism group of at most finitely many finite groups

History
The result was proved by Hariharan K. Iyer in his paper On Solving the Equation Aut(X) = G in the Rocky Mountain Journal of Mathematics, Volume 9, No. 4, Fall 1979.

Statement
Let $$G$$ be a finite group. Then, there are at most finitely many finite groups $$H$$ such that $$\operatorname{Aut}(H) = G$$.

Note that:


 * It is possible that there is no finite group whose automorphism group is $$G$$.
 * There may exist infinite groups whose automorphism group is $$G$$. For instance, the automorphism group of the group of integers is the cyclic group of order two.

Related facts

 * Trivial automorphism group implies trivial or order two
 * More facts in Category:Results about control via the automorphism group

Similar and opposite facts about inner and outer automorphism groups

 * Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group (and in fact, of infinitely many finite simple non-abelian groups, since the construction has an arbitrary prime)
 * Every countable group occurs as the outer automorphism group of a finitely generated group