The construction starts with a quintic V with 20 double points.
Let W be the surface obtained by blowing up the 20 double points.
Suppose that W has a double cover X branched over the 20 exceptional −2-curves.
If there is a group of order 5 acting freely on all these surfaces, then the quotient Z of Y by this group of order 5 is a Catanese surface.
Catanese found a 4-dimensional family of curves constructed like this.