Modular curves are the most famous example of the title. As moduli space towers they exhibit a "Frattini property," based on their monodromy groups as covers of the j-line. Using the goals of Serre's "l-adic representations" book I will treat, in parallel, two cases of general ideas.
Modular curves here derive from the semi-direct product of Z/2 acting through
multiplication by -1 on Z; and
the equally rich case from Z/3 acting irreducibly on Z2.
This view has modular curves as families of sphere covers attached to dihedral groups. In this case we see something familiar their cusps and the monodromy on homology in a fiber in a new way. Then, with analogous methods, we outline the 2nd case to show how the tools extend. To take Serre's Open Image Theorem beyond modular curves, to general moduli of abelian varieties, has failed to master the limiting effect of correspondences read motives of arithmetic monodromy on special tower fibers. Our Z/3 case shows how Frattini data in our Hurwitz space approach helps tame that structure.