On Morita equivalences with endopermutation source and isotypies

We introduce a new type of equivalence between blocks of finite group algebras called an almost isotypy. An almost isotypy restricts an isotypy in Brou\'{e}'s original definition, and it is close to Linckelmann's version. We show that a bimodule of two block algebras of finite groups - which has an endopermutation module as a source and which induces a Morita equivalence - gives rise to an almost isotypy if and only if the character values of a (hence any) source are rational integers. This generalises a previous result of Huang and Zhou. Consequently, if two blocks are Morita equivalent via a bimodule with endopermutation source, then they are almostly isotypic.