\textit{D. H. Foster} [J. Math. Anal. Appl. 67, 549--564 (1979; Zbl 0409.22001)] first introduced the notion of fuzzy topological groups. In the present paper, the concept of \(I\)-fuzzy topological groups is introduced and fundamental framework of \(I\)-fuzzy topological groups is established. This notion is a generalization of \(I\)-topological groups. A characterization of products of \(I\)-fuzzy topological groups in terms of \(I\)-fuzzy quasi-coincident neighbourhood systems is obtained. A characterization of \(I\)-fuzzy topological groups in terms of the corresponding \(I\)-fuzzy quasi coincidence neighbourhood systems at \(e_{\lambda}\) (\(\lambda\in [0, 1]\)) is studied. The relation between the category of \(I\)-fuzzy topological groups and the category of antichain \(I\)-topological groups is discussed. It is shown that the category of \(I\)-fuzzy topological groups and fuzzy continuous homomorphisms is isomorphic to the category of antichain \(I\)-topological groups and \(I\)-continuous homomorphisms; and the category of \(I\)-fuzzy topological groups is topological over the category of groups with respect to the forgetful functor. A direction for further study is given by the authors.