Given an algebraic number field $F$, which we may as well take to be Galois over $\mathbb{Q}$, we denote by $S_F$ the set of rational primes that split in $F$. Sets of the form $S_F$ are called Galoisian. At some point, there was a discussion

Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?

of the fact that the abelian Galoisian sets, that is, $S_F$ corresponding to $F$ abelian over $\mathbb{Q}$, are exactly the sets of primes defined by congruence conditions.

A while later, Matthew Emerton gave this nice answer

Galoisian sets of prime numbers

to a question of Chandan Singh Dalawat about non-abelian Galoisian sets.

I made a comment there I thought I would upgrade to a question. As Matthew points out, Neukirch's remark that the Langlands program provides a characterization of all Galoisian sets is probably meant as a metaphor for some other process. However, I couldn't help but hope that the characterization could be taken literally, at least for some special families. For example, we will refer to a number field $F$ as being of $GL_2$ type if it is the fixed field of $Ker(\rho)$, where $$\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{C})$$ is an irreducible two-dimensional Artin representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$*. Now call a set of primes a solvable $GL_2$ Galoisian set if it is of the form $S_F$ for some solvable extension $F$ of $GL_2$-type.

The question then is: can one use the Langlands program (or anything else) to give a sensible characterization of solvable $GL_2$ Galoisian sets?

There is an awkward sort of guess I could make, but I would rather see what the experts say.

*The idea that we should simply organize fields in this manner corresponding to representations is perhaps a valuable perspective coming out of the Langlands program.