A conjecture on B-groups

In this note, I propose the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$ -group $\beta (G)$ is nilpotent. I give a proof of this conjecture under the additional assumption that $G$ be solvable. I also show that this conjecture is equivalent to the following: the kernel of restrictions to nilpotent subgroups is a biset-subfunctor of the Burnside functor.