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The equivalence of principal bundles with transitive Lie groupoids due toEhresmann is a well known result. A remarkable generalisation of thisequivalence, due to Mackenzie, is the equivalence of principal bundleextensions with those transitive Lie groupoids over the total space of aprincipal bundle, which also admit an action of the structure group byautomorphisms. This paper proves the existence of suitably equivarianttransition functions for such groupoids, generalising consequently theclassification of principal bundles by means of their transition functions, toextensions of principal bundles by an equivariant form of \v{C}ech cohomology.Comment: 23 page

Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel