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Let $\ell\geq 5$ be a prime number and $\mathbb{F}_\ell$ denote the finitefield with $\ell$ elements. We show that the number of Galois extensions of therationals with Galois group isomorphic to $GL_2(\mathbb{F}_\ell)$ and absolutediscriminant bounded above by $X$ is asymptotically at least$\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}$. We alsoobtain a similar result for the number of surjective homomorphisms$\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)$ordered by the prime to $\ell$ part of the Artin conductor of $\rho$.

Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$