Semilinear transformations

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic $p$ with the projective general group $\mathrm{PGL}(m,q)$ and the general linear group $\mathrm{GL}(m,q)$ as Galois groups where $m>1$ is any integer and $q>1$ is any power of $p$. These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group $\mathrm{P}\Gamma \mathrm{L}(m,q)$ and the general semilinear group $\Gamma \mathrm{L}(m,q)$. We also obtain the semilinear versions of the local coverings considered in previous papers.