For most fields $K$ and integers $n$, the general linear group $\mathrm{GL}_n(K)$ has very few normal subgroups - they are precisely the subgroups that either contain the commutator subgroup or are included in the center. The only exceptions to this occur when $K$ either has two or three elements, both with $n=2$. In the former case, $\mathrm{GL}_2(\mathbb{F}_2)\simeq S_3$ has the additional normal subgroup $A_3$, while the latter case contains an additional normal $Q_8$ (as the unique $2$-Sylow of $\mathrm{SL}_2(\mathbb{F}_3)$). While it is straightforward to interpret these two exceptional normal subgroups when regarding $\mathrm{GL}_2(K)$ as a plain group, I am fairly clueless if there is some linear algebraic way to "see" them, considering that the typical normal subgroups have such clear interpretations (scalar matrices and kernel of $\det$).
There may be several ways to see this happening. One way might be to exhibit these subgroups as stabilizers of some exceptional multilinear form on $K^2$ (although I checked that this is not the case for bilinear forms). Another way might be to give an "algebraic" map $\mathrm{SL}_2(K)\to K$ that just happens to be a homomorphism for $K$ small enough. Perhaps this could even come from a general map $\mathrm{GL}_2(K)\to \mathrm{Aff}_1(K)$.
Is it possible to view these subgroups in such a way? Are these two incarnations of general "things" in $\mathrm{GL}_2(K)$ that just happen to be normal (or even subgroups, for that matter) for small $K$? Or is their existence completely unrelated to the vector space from which they come?