be a Sylow $p$-subgroup of $G$ and denote by ${\cal F}_{P}(G)$ the Frobenius category of $G$. The objects of ${\cal F}_{P}(G)$ are subgroups of $P$. For $S, \ T \leq P$, $${\rm mor}_{{\cal F}_{P}(G)}(S, \ T) = \{ \varphi \in {\rm Hom}(S, \ T) \ | \ \varphi = {c_{x}}_{|_S} \ ( \exists  x \in G) \},$$ where $c_x$ is the inner automorphism of $G$ induced by $x$.

[([@Okuyama-Watanabe])]{} If $\tilde{P}$ is abelian, then ${\cal F}_{P}(G) = {\cal F}_{P}(N_{G}(\tilde{P})).$

(Naoki Chigira) $\tilde{P} = P \cap O^{p}(G)$ is strongly closed in $P$ in the sense of [@GLS], §16. Hence the proposition follows from [@GLS], Proposition 16.20. $\blacksquare$

[([@Wata2014], Theorem 3)]{} If $\tilde{P}$ is cyclic, then ${\cal F}_{P}(G) = {\cal F}_{P}(N_{G}(P)).$

Set $\tilde{K} = O_{p'}(C_{{G}}(\tilde{P})) = O_{p'}(C_{\tilde{G}}(\tilde{P})) = C_{\tilde{G}}(\tilde{P})_{p'}$. Then $C_{\tilde{G}}(\tilde{P}) =  \tilde{K} \times \tilde{P}$, and $\tilde{K}$ is normal in $N_{G}(\tilde{P})$. Since $C_{P}(\tilde{P})$ is a Sylow $p$-subgroup of $C_{G}(\tilde{P})$, $C_{G}(\tilde{P}) = C_{\tilde{G}}(\tilde{P}) C_{P}(\tilde{P})  = \tilde{K} C_{P}(\tilde{P})$, by Proposition 1, $$N_{G}(\tilde{P}) = C_{G}(\tilde{P})N_{G}(P) = \tilde{K} N_{G}({P}).$$ This implies $${\cal F}_{P}(N_{G}(\tilde{P})) = {\cal F}_{P}(N_{G}(P)).$$ Note that $[P, K]  \cap P = 1$. This and Proposition 1 complete the proof. $\blacksquare$

The above two propositions are generalized in [@Wata2014]) as follows. Let $b$ be a 