Geodesic
Character Correspondences and π-Special Characters in π-Separable Groups
Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 920-937

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Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups. 
Wolf, Thomas R. Character Correspondences and π-Special Characters in π-Separable Groups. Canadian journal of mathematics, Tome 39 (1987) no. 4, pp. 920-937. doi: 10.4153/CJM-1987-046-1
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