I have several things to add to this discussion, although this is not a complete answer.

$\bullet$ 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.)

$\bullet$ If you fix a field $K$ with $Gal(K/\mathbb{Q}) \cong GL_{2}(\mathbb{Z}/p^{2} \mathbb{Z})$ with $p \geq 5$ prime, there is an algorithm to determine if there is an elliptic curve $E/\mathbb{Q}$ with $\mathbb{Q}(E[p^{2}]) = K$. By the Neron-Ogg-Shafarevich criterion, if there is such a curve $E$, the only primes that ramify in $K$ must be $p$ and the primes that divide the conductor, $N(E)$, of $E$. Moreover, all primes dividing $p N(E)$ ramify in $\mathbb{Q}(E[p^{\infty}])$. However, every minimal subfield of $\mathbb{Q}(E[p^{\infty}])$ is contained in $\mathbb{Q}(E[p^{2}])$. It follows that every prime dividing $p N(E)$ ramifies in $K/\mathbb{Q}$, so one can search for all elliptic curves with bad reduction only at primes that ramify in $K$ (by using the modular symbols algorithms to search for elliptic curves with one of the finitely many possible conductors, for example). (One can see that this doesn't work with $p^{2}$ replaced with $p$ by noting that given $E/\mathbb{Q}$, Rubin and Silverberg construct infinitely many $E'$ with $\mathbb{Q}(E[5]) = \mathbb{Q}(E'[5])$.)

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.)