exhibits $\widehat \mathcal{G}$ as a central $A$-extension of $\mathcal{G}$.

Equivalently, such a central extension $\widehat {\mathcal{G}} \to \mathcal{G}$ is a $(\mathbf{B}A)$-principal 2-bundle.

In traditional literature this is mostly considered for Lie groupoids. Specifically, for $A$ a Lie group and for $C(\mathcal{U})$ the Cech groupoid of a good open cover$\mathcal{U}$ of a smooth manifold$X$, morphisms of smooth stacks$X \to \mathbf{B}^2 A$ are equivalently given by actual morphism of Lie groupoids $C(\mathcal{U}) \to \mathbf{B}^2 A$, which are equivalently degree-2 $A$-cocycles in the Cech cohomology of $X$. The corresponding incarnation $\widehat \mathcal{G}$ of the $\mathbf{B}A$-principal 2-bundle classified by this is known as the $A$-bundle gerbe over $C(\mathcal{U})$.