$|C_{G'}(\rho')|$. Let $a$ (resp. $a'$) be a $p$-element of $G$ (resp. $G'$). For $\rho \in C_{G}(a)_{p'}$ and $\rho'  \in C_{G'}(a')_{p'}$, $$\sum_{i= 1}^{k(b)}\varepsilon_i \chi_i (a\rho) \overline{\chi'_i(a'{\rho}')}$$ $$= \sum_{i = 1}^{k(b)}\left( \sum_{ \pi \in \Pi_a } \sum_{j =1}^{l(b_{\pi})}\varepsilon_i d^{\pi}_{\chi_i \varphi_{j}^{(\pi)}} {\varphi}_{j}^{(\pi)} ({\rho}^{c_{\pi a}}) \right)\left( \sum_{\pi' \in \Pi'_{a'}} \sum_{k =1}^{l(b_{\pi'})}\overline{ d^{\pi'}_{\chi'_i  \varphi'^{(\pi')}_{k}}}\overline{{\varphi'}_{k}^{(\pi')} ({\rho}'^{c'_{\pi' a'}})} \right)$$ $$=  \sum_{\pi \in \Pi_a}\sum_{\pi' \in \Pi'_{a'}}\sum_{k =1}^{l(b_{\pi'})}\left( \sum_{j =1}^{l(b_{\pi})}\left( \sum_{i = 1}^{k(b)} \varepsilon_i d^{\pi}_{\chi_i  \varphi_{j}^{(\pi)}}
\overline{d^{\pi'}_{\chi'_i  \varphi'^{(\pi')}_{k}}}\right) \varphi^{(\pi)}_{j}({\rho}^{c_{\pi a}})\right)\overline{{\varphi'}_{k}^{(\pi')} ({\rho}'^{c'_{\pi' a'}})}.$$ Hence from (1) and by [@NT], Theorem 5.4.11 for basic sets (see [@Navarro], Lemma 7.5) $$\sum_{i= 1}^{k(b)}\varepsilon_i \chi_i (a\rho) \overline{\chi'_i(a'{\rho}')}$$ $$= \sum_{\pi \in \Pi_a \cap \Pi'_{a'}}
\sum_{k = 1}^{l(b_{\pi})}\left( \sum_{j= 1}^{l(b_{\pi})} 
 \left( \sum_{i = 1}^{k(b)} d^{\pi}_{\chi_i \varphi_j^{(\pi)}}
\overline{d^{\pi}_{\chi_i  \varphi_k^{(\pi)}}}\right) \varphi_j^{(\pi)}(\rho^{c_{\pi a}}) \right)\overline{ {\varphi'}^{(\pi)}_{k}(\rho'^{c'_{\pi a'}})}$$ $$= \sum_{\pi \in \Pi_a \cap \Pi'_{a'}}
\sum_{k= 1}^{l(b_{\pi})} \left( \sum_{j = 1}^{l(b_{\pi})} 
c_{\varphi_{j}^{(\pi)}\varphi_{k}^{(\pi)} } \varphi_{j}^{(\pi)}(\rho^{c_{\pi a}})
\right)\overline{\varphi'^{(\pi)}_{k}(\rho'^{c'_{\pi a'}})}\big),$$ here $c_{\varphi_j^{(\pi)} \varphi_k^{(\pi)}}$ is a Cartan integer [w.r.t.]{} the basic set $Bs(b_{\pi})$ in (iv). Thus $\displaystyle\sum_{i= 1}^{k(b)}\varepsilon_i \chi_i (a\rho)\overline{\chi'_i(a'{\rho}')}$ is divisible by $|C_{G}(a\rho)|$ in ${\cal O}$, because $\displaystyle \sum_{j = 1}^{l(b_{\pi})} 
c_{\varphi_j^{(\pi)} \varphi_k^{(\pi)}} \varphi_j^{(\pi)}(\rho^{c_{\pi a}}) $ is divisible by $|C_{G}(\pi \rho^{c_{\pi a}})| = |C_{G}(a\rho)|$ in ${\cal O}$. Similarly $\displaystyle\sum_{i= 1}^{k(b)}\varepsilon_i \chi_i (a\rho)\overline{\chi'_i(a'{\rho}')}$ is by $|C_{G'}(a'\rho')|$ in ${\cal O}$. This completes the proof. $\blacksquare$

Let $P$ 