Lifts of Brauer characters in characteristic two

A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $\phi$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $\phi$ is at most $\mathrm{\mid }Q:Q^{\mathrm{\prime }}\mathrm{\mid }$. 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”.