Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{ P}\Omega^\pm_{ 2m}(\mathbb{ F}_{ \ell^s})$, $\mbox{ PSO}^\pm_{ 2m}(\mathbb{ F}_{ \ell^s})$, $\mbox{ PO}_{ 2m}^\pm(\mathbb{ F}_{ \ell^s})$ or $\mbox{ PGO}^\pm_{ 2m}(\mathbb{ F}_{ \ell^s})$ is a Galois group of $\mathbb{ Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of $\mbox{ GL}_{ 2m}(\mathbb{ A}_\mathbb{ Q})$..
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