  We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal
bundle as a quantization of the gauge groupoid of a classical principal bundle.
We show that the gauge group of the noncommutative bundle is isomorphic to the
group of bisections of the bialgebroid, and we give a crossed module structure
for the bisections and the automorphisms of the bialgebroid. Examples
illustrating these constructions include: Galois objects of Taft algebras, a
monopole bundle over a quantum spheres and a not faithfully flat Hopf--Galois
extension of commutative algebras. The latter two examples have in fact a
structure of Hopf algebroid for a suitable invertible antipode.
