Suppose $G$ is a finite group and $A$ is the set of all character values of $G$. By character values, I mean entries of the character table of $G$. Let $\Gamma = \operatorname{Gal}({\mathbb{Q}(A)}/{\mathbb{Q}})$. Then $\Gamma$ has an action on the set of all conjugacy classes of $G$. Considering orbits, we get an equivalence relation on $G$.

There is another equivalence relation on $G$ in which elements $x$ and $y$ are equivalent if and only if the cyclic subgroups generated by $x$ and $y$ are conjugate subgroups of $G$. My question is about the relationship between the classes defined by the two relations. In an exact phrase:

Question: Is it true that both equivalence relations coincide?

Any suggestions, comments or references to this problem are highly appreciated.

Regards, Alireza