Summary: In this paper, we consider injectivity for nominal sets, \(G\)-sets for a subgroup \(G\) of the group \(\operatorname{Perm}(\mathbb{D})\) of permutations on an infinite countable set \(\mathbb{D}\), with finitely supported elements. Also, we study injectivity of nominal sets with respect to monomorphisms whose domains or codomains are single-orbit. Furthermore, we define the category of \(G_C\)-sets, for finite subsets \(C\) of \(\mathbb{D}\), where \(G_C\) is the set of elements of \(G\) fixing the elements of \(C\), and study injectivity in such categories.