The main purpose of this paper is to prove that the extensions of a nilpotent block algebra and its Glauberman correspondent block algebra are Morita equivalent under an additional group-theoretic condition (see Theorem 1.6); in particular, Harris and Linckelman's theorem and Koshitani and Michler's theorem are covered (see Theorems 7.5 and 7.6). The ingredients to carry out our purpose are the two main results in Külshammer and Puig's work Extensions of nilpotent blocks; we actually revisited them, giving completely new proofs of both and slightly improving the second...
