The aim of this article is the study of right nuclei of quasigroups with right unit element. We investigate an extension process in this category of quasigroups, which is defined by a slight modification of non-associative Schreier-type extensions of groups or loops. The main results
of the article give characterizations of quasigroup extensions satisfying particular nuclear conditions. We apply these results for constructions of right nuclear quasigroup extensions with right inverse property.