On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras

Let A be a finite group acting on a finite group G via automorphisms. Assume that $$(|A|,|G|)=1$$ . We prove that if $$C_G(A)$$ is a Hall $$pi $$ -subgroup of G, then G has a normal $$pi $$ -complement.