The inner automorphism 2-group of a group is essentially the sub-2-group of the automorphism 2-group of of those automorphisms that are connected by a transformation to the identity: this makes the automorphism necessarily an inner automorphism.
In fact, more precisely the inner automorphism 2-group is the 2-group of these connecting transformations, i.e. it remembers the group element and the inner automorphism that it induces under conjugation.
Let be a group. Write for its delooping.
The inner automorphism 2-group of is the strict 2-group
whose objects are diagrams
morphisms are commuting triangles of transformations
Equivalently, this is the action groupoid
of acting on itself.
Equivalently , this is the strict 2-group corresponding to the crossed module
with action given by right multiplication in .
This makes it evident that is contractible
In fact, we may think of as the universal -principal bundle in its incarnation in Grpd (as opposed to the more tradition incarnation in Top, to which it is Quillen equivalent by the homotopy hypothesis theorem).
To emphasize this we also write
We have a natural sequence of groupoids
It is an old theorem by Graeme Segal that under nerve followed by geometric realization this maps to the sequence of topological spaces
that is the universal -bundle over the classifying space in its incarnation in Top.
The 2-group structure on is evident, and hence makes the fact evident that the universal -bundle itself carries a group structure, which is compatibel with the group structure, in that the morphism deloops to a morphism
This fact is useful in various applications in nonabelian cohomology.