  We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal
bundle as a quantization of the classical gauge groupoid of a principal bundle.
When the base algebra is in the centre of the total space algebra, the gauge
group of the noncommutative principal bundle is isomorphic to the group of
bisections of the bialgebroid. In particular we consider Galois objects
(non-trivial noncommutative bundles over a point in a sense) for which the
bialgebroid is a Hopf algebra. For these we give a crossed module structure for
the bisections and the automorphisms of the bialgebroid. Examples include
Galois objects of group Hopf algebras and of Taft algebras.
