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Let  G  be a finite  p -solvable group for some prime  p  and let  A  be a subgroup of Aut( G ) with order relatively prime to  G , such that  A  stabilizes a  p -block  b  of  G  and acts trivially on a defect group  P  of  b . All ordinary characters in  b  are  A -stable by a result of Watanabe. Also there is a block  w ( b ) of  C G  ( A ) such that the Glauberman–Isaacs correspondent characters comprise the ordinary irreducible characters of  w ( b ), and there is a Morita equivalence, and hence a perfect isometry, between the blocks  b  and  w ( b ), given by a bimodule  M  with vertex Δ P  and an endo-permutation module as source and which induces the Glauberman–Isaacs correspondence on the character level. This gives a different proof of results of several authors.

A Morita equivalence for blocks of finite p -solvable groups in the Glauberman–Isaacs–Watanabe correspondence context