The aim of this talk is to define an obstruction theory for making equivariant homotopy diagrams more and more highly commutative. A gluing data which we will use to construct topological spaces with specific homotopy types can be obtained when these obstructions vanish. Given a finite group G, we will also construct finite G-CW-complexes homotopy equivalent to a product of spheres with isotropy groups in a given family of subgroups of G. This is a joint work with Özgün Ünlü.