Local triviality of equivariant algebras

We consider a finite algebra A over a commutative ring R. It is assumed that R is an algebra over the ground field k and that a cocommutative Hopf algebra H acts on R and A in a compatible way. This paper answers the question as to when it is possible to find a ring extension R -> R' such that the R'-algebra A circle times(R)R' is isomorphic with A(0)circle times(k)R' for some k-algebra A(0) and the ring R'circle times(R)R(p) is faithfully flat over the local ring R(p) either for a single prime ideal p of R containing no H-stable ideals of R or for all such primes. If k is algebraically closed, it is shown that A has isomorphic reductions modulo any pair of maximal ideals of R with residue field k containing the same H-stable ideals of R.