e(e-1).$$ By [@Dor], Theorem 68.1, $|{\rm IBr}(b(\tilde{G}))| = e < p$. Hence any irreducible Brauer character of $b(\tilde{G})$ is $G$-invariant, hence any $\tilde{\chi}_i$ are $G$-invariant by Lemma 5. On the other hand we see $(1_{G} * \eta_{\hat{\nu}})\downarrow_{\tilde{G}} =  1_{\tilde{G}} * \eta_{{\nu}}$ (cf. Cabanes [@Cabanes], Theorem 1), hence Lemma 6 and the above imply (5). This completes the proof. $\blacksquare$

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We call $\chi_{i}$ (resp. ${\chi}_{\nu})$ in the above proposition the $canonical$ $extension$ of $\tilde{\chi}_i$ (resp. $\tilde{\chi}_{\nu})$. We set $\chi_1 = 1_{G}$ and call it the canonical extension of $\tilde{\chi}_1 = 1_{\tilde{G}}$. Since $\tilde{\chi}_i$ is $p$-rational from (4), $\chi_i$ is also $p$-rational by Proposition 3 and the uniqueness of $\chi_i$.

Let $G_1$ be a subgroup of $G$ containing $\tilde{G}$ and let $P_1 = P \cap G_1 = \tilde{P} \rtimes (C_{P}(T) \cap G_1)$. Then $P_1$ is a Sylow $p$-subgroup of $G_1$, $\tilde{P}$ is a $p$-hyperfocal subgroup of $G_1$ and $G_1$ also has the inertial index $e$. Let $T$ be a $p$-complement of $N$. Then $P_1$ is normalized by $T$. From (3) $$N_{G_1}(P_1) = O_{p'}(C_{{G_1}}(P_1))TP_1.$$ That is, $O_{p'}(C_{G_1}(P_1))T$ is a $p$-complement of $N_{G_1}(P_1)$ because $T \cap C_{G_1}(P_1) = O_{p'}(C_{G}(P))$. Moreover, for $\nu \in {\rm Irr}_{P}(\tilde{P})$, $$1_{G_1}* \eta_{(\hat{\nu}\downarrow_{P_{1}})} = (1_{G}* \eta_{\hat{\nu}})\downarrow_{G_1}.$$ Therefore the canonical 