is the delooping of an ∞-group$Z(G)$. This is the center of $G$.

Examples

Centers of ordinary groups

For $\mathcal{X} =$∞Grpd and $G$0-truncated, it is an ordinary discrete group. Its automorphism 2-group is the strict 2-group coming from the crossed module$[G \stackrel{Ad}{\to} Aut(G)]$. The morphism $AUT(G) \to Out(G)$ is a fibration hence its homotopy fiber is, up to equivalence, the ordinary fiber, which is the crossed module $(G \stackrel{Ad}{\to} Inn(G))$, where $Inn(G) \subset Aut(G)$ is the group of inner automorphisms. This is equivalent to $(Z(G) \to 1)$, where $Z(G)$ is the ordinary center of $G$, and this is the crossed module corresponding to $\mathbf{B}Z(G)$.