Extensions of $\mathrm{\infty }$-group sheaves

Let $\mathcal{X}$ be an $\mathrm{\infty }$-topos, for example the $\mathrm{\infty }$-category of simplicial sheaves on a Grothendieck site. Then $\mathrm{\infty }$-group sheaves are group objects in $\mathcal{X}$. Let $A\in \mathrm{G}\mathrm{r}\mathrm{p}\mathcal{X}$ be such a group object. Then as $\mathcal{X}$ is an $\mathrm{\infty }$-topos, there exists a universal $\mathbf{B}A$-fiber bundle $\mathbf BA//$\mathbf{A}\mathbf{u}\mathbf{t}$ A\xrightarrow q\mathbf B$\mathbf{A}\mathbf{u}\mathbf{t}$ A$\mathrm{.}Wemake$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 $\mathbf{B}A\mathrm{/}\mathrm{/}\mathbf{A}\mathbf{u}\mathbf{t}A$.