  A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite
$p$-solvable group and $\varphi$ is an irreducible $p$-Brauer character of $G$
with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$.
This conjecture is now known to be true in several situations for $p$ odd, but
there has been little progress for $p$ even. The main obstacle appeared in
characteristic two is that all the vertex pairs of a lift are neither linear
nor conjugate. In this paper we show that if $\chi$ is a lift of an irreducible
$2$-Brauer character in a solvable group, then $\chi$ has a linear Navarro
vertex if and only if all the vertex pairs of $\chi$ are linear, and in that
case all of the twisted vertices of $\chi$ are conjugate. Our result can also
be used to study other lifting problems of Brauer characters in characteristic
two. As an application, we prove a weaker form of Cossey's conjecture for $p=2$
"one vertex at a time".
