A note on the Glauberman correspondence

In this paper, we will investigate the Glauberman-Watanabe correspondence between the characters and blocks. Motivated by the work of Puig and Zhou 10 Puig, L., Zhou, Y. (2012). Glauberman correspondents and extensions of nilpotent block algebras. J. London Math. Soc. (2) 85809–837.Crossref] , Google Scholar]] which is about the extension of a nilpotent block and its Glauberman correspondent, we will modify the basic Morita equivalence obtained in their work and get a new equivalence such that the correspondence between characters induced by this equivalence coincides with the Glauberman correspondence under the situation that the block is a p-extension of a nilpotent block.