  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 $2\notin\pi$. In this paper we introduce the notion of
the twisted vertices in the case where $2\notin\pi$, 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.
