Generating function on epimorphisms between $2$-bridge knot groups

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking account into genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots uknotting number one.