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There is no complete classification, but some structural results are known. To give you something to search for: such groups are called $\mathbb{Q}$-groups. There is a whole book devoted to their structure: Structure and Representations of $\mathbb{Q}$-Groups by F. Kletzing. You will find there answers to many, if not all, of your questions. In particular, the converse of the lemma is indeed true, and is easy to deduce from the fact that under the hypotheses of the lemma, for every irreducible character $\chi$, the values $\chi(g)$ and $\chi(g^m)$ are Galois conjugates.