Conjugacy classes of two element in a group with cyclic Sylow subgroup

Question

Let $G$ be a finite group such that Sylow $p$-subgroup $G$ has order $p$. Let $x$ and $y$ be two elements of order $p$. Is true that $x$ and $y$ are conjugate in $G$?

Answer 1

This is not true. The smallest counterexample is a cyclic group of three elements. For example in the group $\Bbb{Z}/3\Bbb{Z}$ the cosets $\overline{1}$ and $\overline{2}$ are both of order $3$, but they are not conjugate.