\in {\rm Irr}_P(\tilde{P}))$$ where $\chi_i$(resp. $\chi_{\nu}$) is an extension of $\tilde{\chi}_i$ (resp. $\tilde{\chi}_{\nu})$ to $G$.

Since $b(\tilde{G})$ has a cyclic defect group $\tilde{P}$ and since $C_{\tilde{G}}(u)$ is $p$-nilpotent for any $u \in \tilde{P} \backslash \{ 1 \}$, $$(*) \ \ \ (1_{\tilde{G}}^{(1, b(\tilde{G}))}, 1_{\tilde{G}}^{(1, b(\tilde{G}))})  = 1 - \sum_{u}(1_{C_{\tilde{G}}(u)}^{(u, b(C_{\tilde{G}}(u))}, 1_{C_{\tilde{G}}(u)}^{(u, b(C_{\tilde{G}}(u))})$$ $$= 1 - \frac{p^{n} -1}{e}\frac{1}{p^n} = \frac{(e-1)p^n + 1}{ep^{n}},$$ where $u$ runs over a set of representatives for the conjugacy classes of the $p$-elements of $\tilde{G}$. Hence we have $$(1_G^{(1, b(G))}, 1_G^{(1, b(G))}) = (1_G^{(1, b(\tilde{G}))}, 1_G^{(1, b(\tilde{G}))})\frac{1}{|G:\tilde{G}|} = \frac{(e-1)p^n + 1}{e|P|}.$$ Let $\Pi$ be a set of representatives for the $N$-conjugacy classes of $P \backslash \{ 1 \}$. By Proposition 2, $\Pi$ is a set of representatives for the conjugacy classes of $p$-elements of $G$. Let $\Pi_1 = \{ u \in \Pi \ | \  e_u = 1 \}$ and $\Pi_2 = \Pi \backslash \Pi_1 =  \{ u \in \Pi \ | \ e_u = e \}$. Set $b_u = b(C_{G}(u))$ for $u \in \Pi$ and $U = \bigcup_{u \in \Pi_2} u^P$. By Lemmas 1 and 2, $$\sum_{u \in \Pi_1}(1_{G}^{(u, b_u)}, 1_{G}^{(u, b_u)})  =  \sum_{u \in \Pi_1}|C_{P}(u)|^{-1}$$ $$= \frac{1}{e|P|} \sum_{u \in \Pi_1} e |P: C_{P}(u)| = \frac{1}{e|P|}\big(|P| 