The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well-known result. A remarkable generalization of this equivalence, given by Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. In this paper the existence of suitably equivariant transition functions is proved for such groupoids, generalizing consequently the classification of principal bundles by means of their transition functions, to extensions of principal bundles by an equivariant form of Čech cohomology.

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