Groupoid extensions, principal 2-group bundles and characteristic classes

We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]bundles over Lie groupoids (i.e. [G → Aut(G)]-bundles over differentiable stacks) and, on the other hand, centralG-extensions of Lie groupoids (i.e. Ggerbes over differentiable stacks). We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we prove that the universal characteristic classes coincide with the Diximer Douady classes that can be computed from connection-type data.