  We give a class of examples of vector bundles on a relative smooth projective
curve over Spec Z such that for infinitely many prime reductions the bundle has
a Frobenius descent, but the restriction to the generic fiber in characteristic
zero is not semistable. In the third section of the paper we prove for a large
class of varieties (including abelian varieties) that any vector bundle with
this Frobenius descent property is generically semistable.
