On fibrations of Lie groupoids

As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber $\mathsf{A}$, i.e., $$1\to \mathsf{A}\to \mathsf{G}\to \mathsf{K}\to 1$$ where $\mathsf{A},\mathsf{G}$ and $\mathsf{K}$ are Lie groupoids. Similar to the theory of group extensions, we show that the existence of locally topological product Lie groupoid fibrations with fiber $\mathsf{A}$ over $\mathsf{K}$ is obstructed by a groupoid cohomology of $H^3_{\bar \Lambda}({\sf K},Z_{\sf A})$, and these locally topological product Lie groupoid fibrations are classified by $H_{\bar{\mathrm{\Lambda }}}^2(\mathsf{K},Z_{\mathsf{A}})$ once exists. Here $Z_{\mathsf{A}}$ is the center of $\mathsf{A}$. This generalizes the theory of group extensions, of gerbes over manifolds/groupoids and etc.