Vertices for characters of $p$-solvable groups

Suppose that $G$ is a finite $p$-solvable group. We associate to every irreducible complex character $\chi \in \operatorname {Irr}(G)$ of $G$ a canonical pair $(Q,\delta )$, where $Q$ is a $p$-subgroup of $G$ and $\delta \in \operatorname {Irr}(Q)$, uniquely determined by $\chi$ up to $G$-conjugacy. This pair behaves as a Green vertex and partitions $\operatorname {Irr}(G)$ into “families" of characters. Using the pair $(Q, \delta )$, we give a canonical choice of a certain $p$-radical subgroup $R$ of $G$ and a character $\eta \in \operatorname {Irr}(R)$ associated to $\chi$ which was predicted by some conjecture of G. R. Robinson.