  Let $A \subset H$ be a finite centralizing extension of noetherian Hopf
algebras, and let $X$ denote the group of characters of $H$ that restrict to
the augmentation map on $A$. Our main results provide necessary and sufficient
conditions for the fibers of the canonical surjection from $spec H$ onto $spec
A$ to coincide with the $X$-orbits in $spec H$. In particular, all of the
fibers are $X$-orbits if and only if the fiber over the augmentation ideal of
$A$ is an $X$-orbit.
  An application to the representation theory of quantum function algebras, at
roots of unity, is presented.
