  The equivalence of principal bundles with transitive Lie groupoids due to
Ehresmann is a well known result. A remarkable generalisation of this
equivalence, due to 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. This paper proves the existence of suitably equivariant
transition functions for such groupoids, generalising consequently the
classification of principal bundles by means of their transition functions, to
extensions of principal bundles by an equivariant form of \v{C}ech cohomology.
