Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension 1\to Z\to \hat{K}\to K\to 1 of K. It is a classical question whether there exists a \hat{K}-principal bundle \hat{P} on M such that \hat{P}/Z is isomorphic to P. Neeb defines in this context a crossed module of topological Lie algebras whose cohomology class [\omega_{\rm top alg}] is an obstruction to the existence of \hat{P}. In the present paper, we show that [\omega_{\rm top alg}] is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.