Morita equivalent blocks in non-normal subgroups and p-radical blocks in finite groups

Let O be a complete discrete valuation ring with unique maximal ideal J(O), let K be its quotient field of characteristic 0, and let k be its residue field O/J(O) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the [GI-th roots of unity, where /G/ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose: that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R is an element of{O,k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with 'naturally Morita equivalent blocks A and B', which means that A is isomorphic to a full matrix algebra of B, as studied by B. Kulshammer. However, Kulshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Kulshammer. Moreover, in the case His normal in G, we get the same results as Kulshammer; however, he uses the results of E. C. Dade, and we do not.