Monomiality lifts along a quotient

Choose a representation $\rho$ affording $\chi$. By [F1], $N\subseteq \ker \rho$, so [F2] factors $\rho$ through an irreducible representation $\bar{\rho }$ of $Q:=G\mathrm{/}N$. Its character $\bar{\chi }$ satisfies $\chi (g)=\bar{\chi }(gN)$. By the hypothesis and [F3], choose $\bar{H}\le Q$ and a one-dimensional $\bar{H}$-module $W$ with character $\bar{\lambda }$ such that $\bar{\chi }={\mathrm{Ind}}_{\bar{H}}^Q\bar{\lambda }$.

Let $\pi :G\to Q$ be the quotient map and $H={\pi }^{-1}(\bar{H})$. Then $N\le H\le G$ and $H\mathrm{/}N=\bar{H}$. Make $W$ an $H$-module by $h\cdot w:=(hN)\cdot w$; its linear character is $\lambda (h)=\bar{\lambda }(hN)$. In particular, $N$ acts trivially on $W$.

For $\bar{f}\in {\mathrm{Ind}}_{\bar{H}}^{Q}W$, define $\mathrm{\Phi }(\bar{f})(g):=\bar{f}(gN)$. If $h\in H$, then [F4] and [F5] give $\mathrm{\Phi }(\bar{f})(gh)=\bar{f}((gN)(hN))=(hN{)}^{-1}\cdot \bar{f}(gN)=h^{-1}\cdot \mathrm{\Phi }(\bar{f})(g)$. Thus $\mathrm{\Phi }(\bar{f})\in {\mathrm{Ind}}_H^{G}W$, and $\mathrm{\Phi }$ is linear.

Conversely, for $F\in {\mathrm{Ind}}_H^{G}W$ and $n\in N$, [F4] gives $F(gn)=n^{-1}\cdot F(g)=F(g)$ by step 2.1. Hence $\bar{f}(gN):=F(g)$ is well defined; the same covariance calculation shows that $\bar{f}\in {\mathrm{Ind}}_{\bar{H}}^{Q}W$. This construction is inverse to $\mathrm{\Phi }$, so $\mathrm{\Phi }$ is a linear bijection.

For $x,g\in G$, the action in [F4] gives $\mathrm{\Phi }((xN)\cdot \bar{f})(g)=\bar{f}((x^{-1}g)N)=(x\cdot \mathrm{\Phi }(\bar{f}))(g)$. Thus $\mathrm{\Phi }$ is $G$-equivariant when the $Q$-module on the left is inflated along $\pi$.

Taking characters of the isomorphic $G$-modules in steps 4.1–4.2, [F4] gives $\chi (g)=\bar{\chi }(gN)={\mathrm{Ind}}_H^G\lambda (g)$ for every $g\in G$. By [F3], $\chi$ is monomial.