$q$-\'etale covers of cyclic $p$-gonal covers

In this paper we study the Galois group of the Galois cover of the composition of a $q$-cyclic 'etale cover and a cyclic $p$-gonal cover for any odd prime $p$. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.