On two notions of a Gerbe over a stack

Abstract: Let $\mathcal{G}$ be a Lie groupoid. The category $B\mathcal{G}$ of principal $\mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $\mathcal{D}$, there exists a Lie groupoid $\mathcal{H}$ such that $B\mathcal{H}$ is isomorphic to $\mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $F\colon \mathcal{D}\rightarrow \mathcal{C}$, such that $F$ and the diagonal map $\Delta_F\colon \mathcal{D}\rightarrow \mathcal{D}\times_{\mathcal{C}}\mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.