Galois Realisations of ${\mathrm{PSL}}_2({\mathbb{F}}_{p^2})$ via non-unirational Hilbert Irreducibility

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups ${\mathrm{PSL}}_2({\mathbb{F}}_{p^2})$ subject to congruence conditions on $p$.