Let $H$ be a Hall subgroup of finite group $G$. If $H\leq Z(N_{G}(H))$ then $H$ has a normal complement in $G$.
I want to know if we assume $|H|=n $, is $H$ the only subgroup of $ N_{G}(H)$ of order $ n$?
Or is the below statement true for Hall subgroups?
"Let $H$ be a Hall subgroup of the finite group $G$. Also assume that $a,b \in C_{G}(H)$. If $a,b$ are conjugate in $G$, then they are conjugate in $ N_{G}(H)$."
If one of the two is true I can answer the main question. Otherwise, how can I solve the main question?