The uniqueness of vertex pairs in $\pi$-separable groups

Let $G$ be a finite $\pi$-separable group, where $\pi$ is a set of primes, and let $\chi$ be an irreducible complex character that is a $\pi$-lift of some $\pi$-partial character of $G$.It was proved by Cossey and Lewis that all of the vertex pairs for $\chi$ are linear and conjugate in $G$ if $2\in \pi$, but the result can fail for $2\mathrm{\notin }\pi$. In this paper we introduce the notion of the twisted vertices in the case where $2\mathrm{\notin }\pi$, and establish the uniqueness for linear twisted vertices under the conditions that either $\chi$ is an $\mathcal{N}$-lift for a $\pi$-chain $\mathcal{N}$ of $G$ or it has a linear Navarro vertex, thus answering a question proposed by them.