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It is an open problem as to whether any bimodule inducing a Moritaauto-equivalence of a block must have endopermutation source. We prove that,for blocks $b$ with normal defect groups in odd characteristic, a strongerresult holds, namely that all such bimodules have linear source. We also provethe analogous result in characteristic $2$, provided that the defect group isof a specific, slightly restrictive, form.

Picard groups for blocks with normal defect groups and linear source bimodules