A finite group $G$ is called *rational* if every element $g \in G$ is conjugate to all of its primitive powers $g^a, a \in (\mathbb{Z}/\operatorname{order}(g))^\times$.
Analogously, I'll call $G$ *real* if every $g$ is conjugate to its inverse. The ratio of these notions is something I'll call *rational-relative-to-real* (RRR): for every $g$, every primitive power $g^a$ should be conjugate to either $g$ or $g^{-1}$ (or both).

These conditions arise naturally in character theory. Given a field $\mathbb{k}$, let me write $\mathrm{R}_{\mathbb{k}}(G)$ for the ring of $\mathbb{k}$-linear representations of $G$. So $\mathrm{R}_{\mathbb{C}}(G) = \mathrm{RU}(G) = \mathrm{R}(G) $ and $\mathrm{R}_{\mathbb{R}}(G) = \mathrm{RO}(G)$. Then $G$ is rational when $$\mathrm{R}_{\mathbb{Q}}(G) \otimes \mathbb{Q} \to \mathrm{R}_{\mathbb{C}}(G) \otimes \mathbb{Q}$$ is an isomorphism; $G$ is real when $$\mathrm{R}_{\mathbb{R}}(G) \otimes \mathbb{Q} \to \mathrm{R}_{\mathbb{C}}(G) \otimes \mathbb{Q}$$ is an isomorphism; and $G$ is RRR when $$\mathrm{R}_{\mathbb{Q}}(G) \otimes \mathbb{Q} \to \mathrm{R}_{\mathbb{R}}(G) \otimes \mathbb{Q}$$ is an isomorphism.

According to Geoff Robinson's answer to a previous question, finite groups are almost never rational: any abelian composition factor has order at most 11; other than alternating groups, there are only five nonabelian simple groups that can appear.

The question I most care about it:

Which finite simple groups are RRR?

For example, which sporadic groups are RRR?

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