${\mathrm{B}}_{\pi }$-characters and quotients

Let $\pi$ be a set of primes, and let $G$ be a finite $\pi$-separable group. We consider the Isaacs ${\mathrm{B}}_{\pi }$-characters. We show that if $N$ is a normal subgroup of $G$, then ${\rm B}\pi (G/N) = {\rm Irr} (G/N) \cap {\rm B}\pi (G)$.