Lower bounds for the number of number fields with Galois group $G{L}_2({\mathbb{F}}_{\mathrm{\ell }})$

Let $\mathrm{\ell }\ge 5$ be a prime number and ${\mathbb{F}}_{\mathrm{\ell }}$ denote the finite field with $\mathrm{\ell }$ elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to $G{L}_2({\mathbb{F}}_{\mathrm{\ell }})$ and absolute discriminant bounded above by $X$ is asymptotically at least $\frac{X^{\frac{\mathrm{\ell }}{12(\mathrm{\ell }-1)\mathrm{\#}G{L}_2({\mathbb{F}}_{\mathrm{\ell }})}}}{\log X}$. We also obtain a similar result for the number of surjective homomorphisms $\rho :Gal(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to G{L}_2({\mathbb{F}}_{\mathrm{\ell }})$ ordered by the prime to $\mathrm{\ell }$ part of the Artin conductor of $\rho$.