matrix of generalized decomposition numbers of $b(G)$ with respect to a suitable basic set $\{ \varphi_1^{(u)} = 1_{C_{G}(u)}, \varphi_2^{(u)}, \cdots, \varphi_e^{(u)} \}$ for $b_u$ is of the form : $$\left.\begin{array}{|c|cccc|c}
\hline
 & \varphi_1^{(u)} = 1_{C_{G}(u)}&\varphi_2^{(u)} &\cdots &\varphi_e^{(u)} \\
\hline
\chi_{1}\lambda_1&\varepsilon_{1}\lambda_1(u)& 0&\cdots &0  \\
\chi_{2}\lambda_2&0&\varepsilon_{2}\lambda_2(u)& 0& 0\\
 \vdots&\vdots &\vdots &\ddots & \vdots  \\
\chi_{e}\lambda_e& 0& 0&\cdots&\ \varepsilon_e \lambda_e(u) \\ 
(\chi_{\nu }\lambda_{\nu})\!\uparrow^G_{G_{\nu}}&
\varepsilon ({\lambda_{\nu}}\!\uparrow^{P}_{P_{\nu}})(u)&\varepsilon ({\lambda_{\nu}}\!\uparrow_{P_{\nu}}^{P})(u) &\cdots &\varepsilon ({\lambda_{\nu}}\!\uparrow^{P}_{P_{\nu}})(u) \\
\hline
\end{array} \right.$$ where $\lambda_i \in {\rm Irr}(P/\tilde{P}) = {\rm Irr}(G/\tilde{G})$, $\nu \in {\cal X}$ and $\lambda_{\nu} \in {\rm Irr}(P_{\nu}/\tilde{P}) = {\rm Irr}(G_{\nu}/\tilde{G})$.

By Lemmas 2 and 3, $P_u = C_{P}(u)$ is a Sylow $p$-subgroup of $C_{G}(u)$, $u \in C_{P}(T)$ for some $p$-complement $T$ of $N$ and $C_{\tilde{P}}(u)$ is a $p$-hyperfocal subgroup of $C_{G}(u)$. Moreover $l(b_u) = e$ as is stated in the above. Set $p^{n'} = |C_{\tilde{P}}(u)|$ and $m = \frac{p^{n'} -1}{e}$. Now for $\nu \in {\cal X}$, the stabilizer $P_{\nu}$ of $\nu$ in $P$ is normalized by $N$. If $u \not\in  P_{\nu}$, $\eta_{\nu}\!\!\uparrow^P_{P_{\nu}}\!\!(u) = 0$ and $\chi_{\nu}\!\!\uparrow^G_{G_{\nu}}\!\!(u \rho) = 0$ for any $\rho \in C_{G}(u)_{p'}$. Recall that if $u \in P_{\nu}$, then $\eta_{\hat{\nu}}(u) = e$ (see (6) for $G_1 = G_{\nu}$). So from (7) in Proposition 4, in order to establish 