The monoid Cb of name substitutions originated by Pitts in name abstraction, and the notion of a finitely supported Cb-set appeared in the study of models of homotopy type theory in the works of Gabbay and Pitts. On the other hand, retracts and retractions play a crucial role in most branches of mathematics as well as in computer science where partial morphisms need to be completed. Retracts are the subobjects whose related inclusion morphism have a left inverse, called retraction. In this paper, we study the retracts and retractions of cyclic  finitely supported Cb-sets. We _nd the general de_nition of retractions from a cyclic Cb-set, and give necessary conditions under which retractions exist. Also, fix-simple retracts of a cyclic Cb-set are characterized. Further, the cyclic  finitely supported Cb-sets all whose subobjects are retract, are studied. In particular, we give a necessary condition for a cyclic finitely supported Cb-set to be retractable