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