It is known that ${\rm PSL}_{2}(\mathbb{F})$ can be realized for some $\mathbb{F}$ but definitely not all at present. According to David Zywina's note here, ${\rm PSL}_{2}(\mathbb{F}_{27})$ is the smallest non-abelian finite simple group for which it's not yet known if it occurs as a Galois group over $\mathbb{Q}$.
Jeremy Rouse
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