On the depth 2 condition for group algebra and Hopf algebra extensions

Let R≠0 be a commutative ring, and let H be a subgroup of finite index in a group G. We prove that the group ring RG is a ring extension of the group ring RH of depth two if and only if H is a normal subgroup of G. We also show that, under suitable additional hypotheses, an analogous result holds for extensions of Hopf algebras over R.