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 $Aut(H \times K) =Aut(H) \times Aut(K)$.