A finite group $G$ is called rational if and only if $N_G(\langle x\rangle)/C_G(x)\cong Aut(\langle x\rangle)$ for all $x\in G$. The word ``rational" is because there is an equivalent definition in group representation theoretic terms: A finite group $G$ is rational if and only if for any complex irreducible character $\chi$ of $G$, $\chi(g)$ is rational for all $g\in G$.

Is it true that a finite group $G$ is rational if $N_G(\langle x\rangle)/C_G(x)\cong Aut(\langle x\rangle)$ for any $p$-element $x$ of $G$ and for all primes $p$ dividing $|G|$.