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We show that a bimodule between block algebras which has a fusion stableendopermutation module as a source and which induces Morita equivalencesbetween centralisers of nontrivial subgroups of a defect group induces a stableequivalence of Morita type; this is a converse to a theorem of Puig. Thespecial case where the source is trivial has long been known by many authors.The earliest instance for a result deducing a stable equivalence of Morita typefrom local Morita equivalences with possibly nontrivial endopermutation sourceis due to Puig, in the context of blocks with abelian defect groups with aFrobenius inertial quotient. The present note is motivated by an application,due to Biland, to blocks of finite groups with structural properties known tohold for hypothetical minimal counterexamples to the $Z_p^*$-Theorem.

On stable equivalences with endopermutation source