A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is abelian. Moreover, if the group $G$ is not abelian, those representations which *are* linear can be described by replacing $G$ by $G/G'$ (where $G'$ is the commutator subgroup.)

I need a similar classification in which the field of complex numbers is replaced by the quaternion division ring. Is there a similar theory? **Can I classify those $G$ for which the quaternionic representations are linear?** (Is it possible to even identify a normal subgroup $K$ of $G$ for which I am guaranteed that $G/K$ is "quaternionic linear"?)

I've identified some articles on "quaternionic" or "symplectic" representations but most of these are concerned with infinite groups and assume quite a lot of theory that does not seem relevant to finite groups.

(This mathoverflow question is similar but in that post $G$ is infinite and there is not the emphasis on "all representations are degree 1".)