Outer automorphisms of a holomorph

Question

Let $G$ be a finite group. Although $G$ may have outer automorphisms, an easy way to "kill" them is taking the holomorph, $G\rtimes\mathrm{Aut}(G)$. However, this process may a priori produce more outer automorphisms.

Does $G\rtimes \mathrm{Aut}(G)$ have outer automorphisms?

Surely there is an example of a group $G$ such that $G\rtimes\mathrm{Aut}(G)$ has outer automorphisms, but none come to mind.

Answer 1

The holomorph of the symmetric group $S_3$ is isomorphic to $S_3 \times S_3$. The automorphism of the latter interchanging the two copies of $S_3$ is outer.