G-gerbes, principal 2-group bundles and characteristic classes

Let G be a Lie group and G → Aut(G) be the canonical group homomorphism induced by the adjoint action of a group on itself. 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 and, on the other hand, G-extensions of Lie groupoids (i.e. between principal [G→Aut(G)]- bundles over dierentiable stacks and G-gerbes over dierentiable stacks). This approach also allows us to identify G-bound gerbes and [Z(G) → 1]-group bundles over dierentiable stacks, where Z(G) is the center of G. We also introduce universal characteristic classes for 2-group bundles. For groupoid central G-extensions, we introduce Dixmier-Douady classes that can be computed from connection-type data generalizing the ones for bundle gerbes. We prove that these classes coincide with universal characteristic classes. As a corollary, we obtain further that Dixmier-Douady classes are integral.