A stronger form of Neumann’s BFC-theorem

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing element $x$. A famous theorem B. H. Neumann states that if is in which all classes are finite with bounded size, then derived $G'$ finite. We establish following result. Let $n$ be positive integer and $K$ subgroup such $|x^G|\leq n$ each $x\in K$. $H=\langle K^G\rangle$ normal closure $K$. Then order $H'$ $n$-bounded. Some corollaries this result also discussed.