Galois conjugate blocks of finite group algebras can be isotypic but not p-permutation equivalent, so not every isotypy lifts to a p-permutation equivalence.
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Isotypic blocks of finite groups algebras that are not $p$-permutation equivalent
Galois conjugate blocks of finite group algebras can be isotypic but not p-permutation equivalent, so not every isotypy lifts to a p-permutation equivalence.