  We provide necessary and sufficient conditions to extend the Hopf-Galois
algebra structure on an algebra R to a generalized ambiskew ring based on R, in
a way such that the added variables for the extension are skew-primitive in an
appropriate sense. We show that the associated Hopf algebra is again a a
generalized ambiskew ring, based on a suitable Hopf algebra H(R). Several
examples are examined, including the Hopf-Galois objects over Uq(sl2).
