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 ass ume 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 splitt ing 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 'natur ally 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, Kulshamm er 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 Hi s normal in G, we get the same results as Kulshammer; however, he uses the results of E. C. Dade, and we do not.