  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. In particular, Harris
and Linckelman's theorem and Koshitani and Michler's theorem are covered. The
ingredient to carry out our purpose is the two main results in K\"ulshammer and
Puig's work "Extensions of nilpotent blocks"; we actually revisited them,
giving completely new proofs of both and slightly improving the second one.
