A construction of two distinct canonical sets of lifts of Brauer characters of a p-solvable group

Navarro defined the set $Irr(G\mid Q,\delta )\subseteq Irr(G)$, where $Q$ is a $p$-subgroup of a $p$-solvable group $G$, and shows that if $\delta$ is the trivial character of $Q$, then $Irr(G\mid Q,\delta )$ provides a set of canonical lifts of ${\text{IBr}}_p(G)$, the irreducible Brauer characters with vertex $Q$. Previously, Isaacs defined a canonical set of lifts $\text{bpig}$ of $\text{ipig}$. Both of these results extend the Fong-Swan Theorem to $\pi$-separable groups, and both construct canonical sets of lifts of the generalized Brauer characters. It is known that in the case that $2\in \pi$, or if $\mathrm{\mid }G\mathrm{\mid }$ is odd, we have $\text{bpig}=Irr(G\mid Q,1_Q)$. In this note we give a counterexample to show that this is not the case when $2\in ̸\pi$. It is known that if $N \nrml G$and$\chi \in \bpig$,thentheconstituentsof$\chi_N$arein$\bpi(N)$\mathrm{.}However,weusethesamecounterexampletoshowthatif$N \nrml G$, and $\chi \in Irr(G\mid Q,1_Q)$ is such that $\theta \in Irr(N)$ and $[\theta ,{\chi }_N]\mathrm{\ne }0$, then it is not necessarily the case that $\theta \in \textup{Irr}(N)$ inherits this property.