I have two things to add to this discussion.
$\bullet$ For $n = 2$ and $n = 3$, every Galois extension of $\mathbb{Q}$ with Galois group ${\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ does arise from an elliptic curve (by a result of Shepard-Barron and Taylor from 1997 - see the reference in the paper of Dieulefait linked to below.) For $n = 4$, this is not true (see this paper).
$\bullet$ For $p \geq 7$ prime, it is known that not every Galois representation $\rho : G_{\mathbb{Q}} \to GL_{2}(\mathbb{F}_{p})$ arises from an elliptic curve, even if we restrict $\rho$ to have cyclotomic determinant. This is shown in a paper of Dieulefait for $p \geq 7$, because one can construct Galois representations using modular forms of different weights. This somehow doesn't seem quite the same as starting with a field $K/\mathbb{Q}$ with $Gal(K/\mathbb{Q}) \cong GL_{2}(\mathbb{F}_{p})$. (Any two isomorphisms between $Gal(K/\mathbb{Q})$ and $GL_{2}(\mathbb{F}_{p})$ differ by an automorphism of $GL_{2}(\mathbb{F}_{p})$ and I haven't been able to find a convenient reference for what ${\rm Aut}(GL_{2}(\mathbb{F}_{p}))$ is.)