Sylow subgroup of a group generated by two subgroups

Question

Let $G$ be a finite group and $H=\langle A ,B\rangle$ be a subgroup of $G$ generated by subgroups $A$ and $B$ of $G$. Is it true that $H_r=\langle A_r,B_r\rangle$, where $H_r$, $A_r$ and $B_r$ are Sylow subgroups of $H,A,B$ respectively.

Answer 1

No, take $G=H=S_3$, $A=\langle (12)\rangle$, $B=\langle (13)\rangle$. Note: it is true when $H=AB$, see Sylow $p$-subgroup of a direct product is product of Sylow $p$-subgroups of factors