Blocks inequivalent to their Frobenius twists

Let k be an algebraically closed field of characteristic p. We give a general method for producing examples of blocks B of finite group algebras that are not Morita equivalent as k-algebras to the Frobenius twist B(P). Our method produces non-nilpotent blocks having one simple module and elementary abelian defect group. These also provide the first known examples of blocks where there is a perfect isotypy at the level of ordinary characters with all the signs positive, but no derived equivalence between the blocks. We do not know of any examples of blocks B that are not Morita equivalent to the second Frobenius twist B(P-2).