TITLE: Uses of the holomorph, Hol($G$) = $G \rtimes $ Aut($G$)
QUESTION [30 upvotes]: In every group theory textbook I've read, the holomorph has been defined, and maybe a few problems done with it. I've also seen papers focusing on computing Hol($G$) for a specific class of $G$.
One thing I have never seen is any actual use for it. Are there major results using the holomorph of a group? Does it occur in the proof of any useful theorems? 
It seems intrisically interesting to me since it allows you to treat automorphisms of a group and elements of a group uniformly, and I would definitely like to learn more about it.

REPLY [6 votes]: Another extensive use of the holomorph is in the study of crossed modules /  2-groups (in the categorification sense).  Any group yields a one object groupoid. Any groupoid $G$ yields an endomorphism gadget $G^G$, that is the category of functors from $G$ to itself. This is at one and the same time a groupoid and a monoid (under composition of functors). Now look at the functors that are isomorphisms. That gives you a category which is also a group.  (Yes I do mean that! It is the subgroup of the monoid structure.) That makes it a 2-group in that categorified sense.  Any 2-group yield a crossed module and the crossed module here is just the inner automorphism homomorphism $G\to Aut(G)$.  The 2-group is the holomorph!
This basic construction is central to many treatments of non-Abelian cohomology (of groups, and of sheaves of groups.) See Larry Breen's papers, and more recently papers by Aldrovandi and Noohi. (You can find stuff on this in the n-Lab and also in my Menagerie notes, a version of which is on my n-Lab homepage.) It is also central to attempts to implement Grothendieck's Pursuing Stack programme although that needs an enormous amount more input as well!
(I can give detailed references if they are of interest.)