Let pi be a set of primes. The paper studies some special homomorphs of finite pi-solvable groups, proving that some of them are Schunck classes. These homomorphs are used to give conditions on an arbitrary homomorph (X) under bar, such that any Hall pi-subgroup of a finite pi-solvable group G to be an (X) under bar -projector of G. Particularly, for pi the set of all primes, one obtain the converse of a result given by W. Gaschutz in 8.
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