\sum_{\nu \in \tilde{{\cal X}}}\eta_{\nu}(u)\big)\tilde{\chi}_{\mu}(\tau)$$ $$=  1 + \sum_{i =  2}^{e}\varepsilon_i \tilde{\chi}_i(\tau) -  
\varepsilon \tilde{\chi}_{\mu}(\tau).$$ From this $$(1 * \eta_{\mu})(\tau) = e = \big((e-1)1_{\tilde{G}} - \sum_{i = 2}^{e}\varepsilon_i \tilde{\chi}_i (\tau) + \varepsilon\tilde{\chi}_{\mu}\big)(\tau).$$ This completes the proof. $\blacksquare$

\
We denote by Irr$_{P}(\tilde{P})$ the set of $P$-invariant linear characters of $\tilde{P}$. Note that if $\nu \in {\rm Irr}_{P}(\tilde{P})$ then $\nu^{a} \in {\rm Irr}_{P}(\tilde{P})$ for $a \in N$. Let $\nu \in $Irr$_{P}(\tilde{P})$. Then $\nu$ is trivial on $[\tilde{P}, P]$. Since $P/[\tilde{P}, P] = (\tilde{P}/[\tilde{P}, P]) \times (C_{P}(T)[\tilde{P}, P]/[\tilde{P}, P])$ by Lemma 2, there is a unique extension $\hat{\nu}$ of $\nu$ to $P$ such that $C_{P}(T) \subseteq {\rm Ker}\  \hat{\nu}$, where $T$ is a $p$-complement of $N$. In fact $C_{P}(U) \subseteq {\rm Ker}\  \hat{\nu}$ for any $p$-complement $U$ of $N$. We call $\hat{\nu}$ the $canonical$ $extension$ of $\nu$. For $a \in N$, $({\hat{\nu}})^a = \widehat{\nu^{a}}$. We set $$\eta_{\hat{\nu}} = \sum_{a \in E}\hat{\nu}^a.$$ This is $N$-invariant, and hence $\eta_{\hat{\nu}}$ is $G$-stable by Proposition 2. Of course $\eta_{\hat{\nu}}\downarrow_{\tilde{P}} = \eta_{\nu}$. Also note $\eta_{\hat{\nu}}(u) = e$ if $u \in C_{P}(T)$.

Let $\nu \in {\rm Irr}_{P}(\tilde{P})$. Under the above notations, $$1_{{G}} * \eta_{\hat{\nu}} = (e - 1)1_{{G}} - \sum_{i = 2}^{e}\varepsilon_i {\chi}_i + \varepsilon {\chi}_{\nu}, \ (\forall \nu 