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I wanted to ask what is known about finite groups whose automorphism group is decomposable (i.e. $Aut(G)=H \times K$ for some groups $H,K$) ? Is there for instance some kind of classification or do such groups share some interesting properties ?

The only general fact I know about the subject is the well-known fact that if $H,K$ are finite groups of coprime order then $Aut(H \times K) =Aut(H) \times Aut(K)$.

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Consider just the case of cyclic groups. The cyclic groups with this property are those whose order is neither 2 nor a Fermat prime. – Will Sawin Apr 21 '12 at 8:24
More generally, if $H$ and $K$ are groups for which ${\rm Hom}(H,K)$ and ${\rm Hom}(K,H)$ are both trivial then ${\rm Aut}(H \times K) = {\rm Aut}(H) \times {\rm Aut}(K)$. – Derek Holt Apr 21 '12 at 13:20
@Will and Derek: These are interesting examples I wasn't aware of so far. Many thanks. – Todd Leason Apr 21 '12 at 18:37
Also, if $T$ is a finite nonabelian simple group then $Aut(T^n)\cong Aut(T)\wr S_n$. More generally, the structure of $Aut(H_1\times \cdots \times H_n)$ for arbitrary finite groups $H_i$ is described by Jonni Bidwell in his Ph.D. thesis. Though it won't answer your question, you can find this work at: – William DeMeo Apr 25 '12 at 10:37

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