  If a group $G$ is $\pi$-separable, where $\pi$ is a set of primes, the set of
irreducible characters $\operatorname{B}_{\pi}(G) \cup
\operatorname{B}_{\pi'}(G)$ can be defined. In this paper, we prove that there
are variants of some classical theorems in character theory, namely the Theorem
of Ito-Michler and Thompson theorem on character degrees, which involve
irreducible characters in the set $\operatorname{B}_{\pi}(G) \cup
\operatorname{B}_{\pi'}(G)$.
