Let $G$ be a finite group, $A\le H$ where $H$ is a proper subgroup of $ G$.
How to show that if $H$ is permutable in $G$ (i.e. $HB = BH$ for all $B\le G$) then $A$ is permutable in $G$?
Well, I started by assuming the opposite (if $A$ is not permutable then $H$ is not permutable) and got two cases, when $B$ is $H$ and not, but I couldn't continue. Can you give me a hint, please?