  We construct an invertible normalised 2 cocycle on the Ehresmann Schauenburg
bialgebroid of a cleft Hopf Galois extension under the condition that the
corresponding Hopf algebra is cocommutative and the image of the unital cocycle
corresponding to this cleft Hopf Galois extension belongs to the centre of the
coinvariant subalgebra. Moreover, we show that any Ehresmann Schauenburg
bialgebroid of this kind is isomorphic to a 2-cocycle twist of the Ehresmann
Schauenburg bialgebroid corresponding to a Hopf Galois extension without
cocycle, where comodule algebra is an ordinary smash product of the coinvariant
subalgebra and the Hopf algebra (i.e. $\C(B/#_{\sigma}H, H)\simeq \C(B\#H,
H)^{\tilde{\sigma}}$). We also study the theory in the case of a Galois object
where the base is trivial but without requiring the Hopf algebra to be
cocommutative.
