Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\mathbb{Q}_l$).

I know that Serre proved that for $p\geq 11$, $\bar{\rho}_{E,p}(G_p)$ cannot be an exceptional group (i.e. isomorphic to one of $A_4,S_4$ or $A_5$) by showing that inertia is "too big". Is something like this known for $\bar{\rho}_{E,p}(G_l)$ when $l \neq p$? I know that the image of inertia $\bar{\rho}_{E,p}(I_l)$ can only be cyclic of order $2,3,4,6$ for $l>3$, however, this is not very helpful in excluding the exceptional groups. My impression is that this $\bar{\rho}_{E,p}(G_l)$ should not be exceptional as a subgroup of $\text{GL}_2(\mathbb{F}_p)$ in general. Are there any cases known when this statement is false or any reference where it is proven?