For $G$ a group, the outer automorphism group $Out(G)$ is the quotient of the automorphism group by the normal subgroup of inner autormorphisms:
Write
for the automorphism 2-group of $G$. This is the strict 2-group coming from the crossed module
Therefore the 0-truncation of $AUT(G)$ is $Out(G)$:
This perspective generalizes to the notion of outer automorphism ∞-group.
The connected components of the subgroup of outer automorphisms of the a super Poincaré group which fixes the underlying Poincaré group is known as the R-symmetry group in supersymmetry.
outer automorphism group, outer automorphism ∞-group