Epimorphisms from 2-bridge link groups onto Heckoid groups (II)

In Part I of this series of papers, we made Riley's definition of Heckoid groups for 2-bridge links explicit, and gave a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's construction. In this paper, we give a complete characterization of upper-meridian-pair-preserving epimorphisms from 2-bridge link groups onto even Heckoid groups, by proving that they are exactly the epimorphisms obtained by the systematic construction.