Crossed Modules and the Integrability of Lie brackets

We prove that in the transitive case the obstruction to the integrability of a Lie algebroid coincides with the lifting obstruction of a crossed module of Lie groupoids associated naturally with the given Lie algebroid. Then we investigate the generalisation of this result for extensions of transitive Lie algebroids, giving explicitly the corresponding lifting obstruction and classifying the lifts in case the obstruction vanishes