Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}\_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\mathbb{F}\_p$?



Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?
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](http://pi.math.cornell.edu/~zywina/papers/smallGalois.pdf), ${\rm PSL}\_{2}(\mathbb{F}\_{27})$ is the smallest non-abelian finite simple group for which it's not yet known if it occurs as a Galois group over $\mathbb{Q}$.


EDIT: As noted by the OP, David Zywina's [paper](https://pi.math.cornell.edu/~zywina/papers/PSL2Modular.pdf) gives realizations of ${\rm PSL}\_{2}(\mathbb{F}\_{\ell^{3}})$ for $\ell \equiv \pm 2, \pm 3, \pm 4 \text{ or } \pm 6 \pmod{13}$. This still leave the case of ${\rm PSL}\_{2}(\mathbb{F}\_{125})$ unresolved. And now the smallest order finite simple group which is not yet known to occur as a Galois group over $\mathbb{Q}$ is $^{2}B\_{2}(8)$, the smallest Suzuki group, which has order $29120$. (An approach to solving the inverse Galois problem for this group is discussed on page 32 of William Chen's paper [here](https://arxiv.org/abs/1510.05687).)


