We apply the braid orbit theorem to projective semilinear groups over the finite fields with $p^2$ elements and some almost simple groups of Lie type. The projective special linear groups $\textrm{PSL}_2(p^2)$ with $p\equiv\pm 3(\bmod 8)$, the Tits simple group, and some small simple groups occur regularly as Galois groups over the rationals.
