\ b')$-Brauer pair,

[(ii)]{} there exists $\Pi \subseteq P$ such that $\{ (\pi, b_{\pi}) \in (P, b_P) \ | \   \pi \in \Pi \}$ $(resp. \ \{ (\pi, b'_{\pi}) \in (P, b'_P) \ |\  \pi \in \Pi \})$ is a set of representatives for the $G$ $(resp. \ G')$-conjugacy classes of $b$ $(resp. \ b')$- Brauer elements,

[(iii)]{} $l(b_{\pi}) = l(b'_{\pi})$ for any $\pi \in \Pi$,

[(iv)]{} For any $\pi \in \Pi \ \backslash \ \{ 1 \}$, there exist basic set $Bs(b_{\pi}) = \{ \varphi_{j}^{(\pi)} \ | \  j = 1, \ 2, \cdots, l(b_{\pi}) \}$ for $b_{\pi}$ and $Bs(b'_{\pi}) = \{ {\varphi'_j}^{(\pi)} \ | \  j = 1, \ 2, \cdots, l(b'_{\pi}) \}$ for $b'_{\pi}$ such that $$d_{\chi_i \varphi_{j}^{(\pi)}}^{\pi} = \varepsilon_i  d_{\chi'_{i} \varphi_{j}'^{(\pi)}}^{\pi}, \ \varepsilon_i = \pm 1 \ (1 \leq i \leq k(b), \ 
1 \leq j \leq l(b_{\pi}))$$ where [Irr]{}$(b) = \{ \chi_i \ | \ i = 1, 2, \cdots, k(b)\}$ and [Irr]{}$(b') = \{ \chi'_i \ | \ i = 1, 2, \cdots, k(b')\}$. Then $$\chi'_i \in R_{{\cal K}}(G', b') \mapsto \varepsilon_{i} \chi_i \in R_{{\cal K}}(G, b)$$ is a perfect isometry from $b'$ to $b$ in the sense of Broué [@Broue].

By the condition 