Extensions of $\infty$-group sheaves

Abstract: Let $X$ be an $\infty$-topos, for example the $\infty$-category of simplicial sheaves on a Grothendieck site. Then $\infty$-group sheaves are group objects in $X$. Let $A\in Grp(X)$ be such a group object. Then as $X$ is an $\infty$-topos, there exists a universal $BA$-fiber bundle $BA//Aut(A)\xrightarrow q\mathbf B\mathbf{Aut}A$. We make $q$ pointed, and show that as a pointed map, via the looping-delooping equivalence, it is a universal extension of group objects by $A$. In particular, semidirect products of group objects by $A$ are classified by $BA//Aut(A)$.