extension of $\tilde{\chi}_i$ (resp. $\tilde{\chi}_{\nu}$) to $G_1$ coincides with $\chi_{i}\!\! \downarrow_{G_{1}}$ (resp. $\chi_{\nu}\!\! \downarrow_{G_{1}}$).

Let $\nu \in $ [Irr]{}$(\tilde{P})$. We denote by $P_{\nu}$ the stabilizer of $\nu$ in $P$ and set $G_{\nu} = \tilde{G}P_{\nu}$. Since $(\eta_{\nu})^v = \eta_{\nu^v}$ and hence $(1_{\tilde{G}}* \eta_{\nu} )^v = 1_{\tilde{G}}* \eta_{\nu^v}$ for $v \in P$, the equation (5) implies $$(\tilde{\chi}_{\nu})^v = \tilde{\chi}_{\nu^v}.$$ Hence $G_{\nu}$ is the stabilizer of $\tilde{\chi}_{\nu}$ in $G$ because $G = \tilde{G}P$. Since $C_{P}(\tilde{P}) \subseteq P_{\nu}$ and $P/C_{P}(\tilde{P})$ is a cyclic $p$-group isomorphic to a subgroup of Aut$(\tilde{P})$, $P_{\nu}$ is normalized by $N$ and hence $G_{\nu}$ is normal in $G$. Let $\hat{\nu}$ be the canonical extension of $\nu$ to $P_{\nu}$ and let $\chi_{\nu}$ the canonical extension of $\tilde{\chi}_{\nu}$ to $G_{\nu}$. For $\lambda \in {\rm Irr}(P_{\nu}/\tilde{P}$) and for $t \in N_{p'}$, $[P_{\nu}, t] \subseteq [P, t] \subseteq \tilde{P}$ and hence $\lambda^t = \lambda$. Therefore $(\eta_{\hat{\nu}}\lambda)\!\uparrow^{P}_{P_{\nu}}$ is $N$-invariant, and hence $G$-stable. Moreover we note that Irr$(P_{\nu}/\tilde{P})$ can be identified with Irr$(G_{\nu}/\tilde{G})$ through the isomorphism $P_{\nu}/\tilde{P} \cong G_{\nu}/\tilde{G}$.

Let $\nu \in $ [Irr]{}$(\tilde{P})$. With the above notations, $ (1_{G_{\nu}} * \eta_{\hat{\nu}})\lambda = 1_{G_{\nu}} * \eta_{\hat{\nu}}\lambda = (e - 1)\lambda - \displaystyle\sum_{i = 2}^{e}\varepsilon_i ({{\chi}_i}\downarrow_{G_{\nu}})\lambda + \ \varepsilon {\chi}_{\nu} \lambda, $ and hence $$1_{G} * ((\eta_{\hat{\nu}} 