Rings of invariants of modular p-groups which are hypersurfaces

For L a finite non-modular group whose invariants form a polynomial ring and H a subgroup of L containing the derived group of L, Nakajima found necessary and sufficient conditions on H for its invariant ring SH to be a hypersurface. In a crucial step of his proof he showed that if SH is a hypersurface, then between H and L there is a group G with polynomial invariant ring such that SH=SG[b]. For G a finite modular p-group over Fp with polynomial invariant ring and H a subgroup of G containing the derived group of G, we find necessary and sufficient conditions on H to ensure that SH=SG[b].