Let $G$ be a finite group. It is known that the endomorphism ring of an irreducible real representation $\rho: G\to\mathrm{GL}(\Bbb R^d)$ is (isomorphic to) either $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). I realize more and more that there are big gaps in my knowledge with regards to this result.
For example, I do know very little about which groups and which representations will produce what type of endomorphism ring. My ignorance goes so far that I do not even know whether the type of endomorphism ring is essentially a property of the group (independent of the [faithful] representation), or whether it can vary from representation to representation.
Question. Are there group theoretic properties of $G$ by which I can tell whether all (or at least some) irreducible (faithful) representations of $G$ will have endomorphism ring $\Bbb C$ or $\Bbb H$? Can a group have (faithful) irreducible representations with different endomorphism rings?
I could sleep even better if I knew that there exists a complete classification or characterization of the groups with a particular type of endomorphism ring. Of course, classifying finite groups is probably too much to ask for, but I always had the (potentially naive) hope that most representations of most groups have endomorphism ring $\Bbb R$. Can this be made precise?
Question. Do most irreducible representations of most groups have endomorphism ring $\Bbb R$? That is, is it tractable to classify the remaining groups/representations with ring $\Bbb C$ and $\Bbb H$?