One would like to embed an abstract group into a bigger group in which every automorphism of is obtained by restricting (to ) an inner automorphism of that fixes as a subset of . The holomorph is the universal (smallest) solution to this problem.
Each group embeds into the symmetric group on the underlying set of by the left regular representation where . The image is isomorphic to (that is, the left regular representation of a discrete group is faithful). The normalizer of the image of in is called the holomorph.
The holomorph occurs very naturally as the group of arrows of the 2-group (groupoid internal to ).