P/\tilde{P}$ and let $P_{\nu}$ be the stabilizer of $\nu$ in $P$ for $\nu \in {\cal X}$. Moreover we set $\bar{P}_{\nu} = P_{\nu}/\tilde{P}$ and $x_i = \chi_i(u)$. We recall $x_i$ is a rational integer. From (7), it suffices to show in order to establish the proposition $$x_i = \varepsilon_i \ (1 \leq i \leq e).$$ So we can assume that $P = \langle \tilde{P}, u \rangle$, hence $$G = \langle \tilde{G}, u \rangle.$$ From (7) again and [@NT], Theorem 5.4.11, $$\sum_{\lambda \in {\rm Irr}(\bar{P})}|\lambda(u)|^2  + \sum_{i = 2}^{e} \sum_{\lambda \in {\rm Irr}(\bar{P})}{x_i}^2|\lambda(u)|^2$$ $$+ \sum_{\nu}\sum_{\lambda_{\nu} \in {\rm Irr}(\bar{P}_{\nu})}\left|((\eta_{\hat{\nu}}\lambda_{\nu})\!\uparrow_{P_{\nu}}^P\big)(u) - (e -1)(\lambda_{\nu}\!\uparrow^{P}_{P_{\nu}})(u) + \sum_{i = 2}^{e}\varepsilon_{i} x_i (\lambda_{\nu}\!\uparrow^{P}_{P_{\nu}})(u)\right|^2$$ $$= |C_{P}(u)|$$ where $\nu$ runs over a set $\{ \nu \in {\cal X} \ | \ P_{\nu} =  P \}$. Hence $$1+ \sum_{i = 2}^{e} {x_i}^2 + \sum_{ \nu }\left|\eta_{\hat{\nu}}(u) - (e -1) + \sum_{i = 2}^{e}\varepsilon_{i} x_i \right|^2  = |C_{\tilde{P}}(u)|$$ where $\nu$ runs over $\{ \nu \in {\cal X} \ |\  P_{\nu} = P \}$. Set $p^{n'}  = |C_{\tilde{P}}(u)|$. By Brauer’s permutation lemma ([@NT], Lemma 3.2.18, (i)), the numbers of $P$-invariant linear characters of $\tilde{P}$ is equal to $|C_{\tilde{P}}(u)|$. Let ${\cal X}_1 = \{ \nu \in {\cal X} \ | \ \nu \in {\rm 