In Rev. Math. Phys. 4 (1997) 785 we study Hilbert-C* systems ${F,G}$ where the fixed point algebra $A$ has nontrivial center $Z$ and where $A'\cap F=Z$ is satisfied. The corresponding category of all canonical endomorphisms of $A$ contains characteristic mutually isomorphic subcategories of the Doplicher/Roberts-type which are connected with the choice of distinguished $G$-invariant algebraic Hilbert spaces within the corresponding $G$-invariant Hilbert $Z$-modules. We present in this paper the solution of the corresponding inverse problem. More precisely, assuming that the given endomorphism category $T$ of a C*-algebra $A$ with center $Z$ contains a certain subcategory of the DR-type, a Hilbert extension ${F,G}$ of $A$ is constructed such that $T$ is isomorphic to the category of all canonical endomorphisms of $A$ w.r.t. ${F,G}$ and $A'\cap F=Z$. Furthermore, there is a natural equivalence relation between admissible subcategories and it is shown that two admissible subcategories yield $A$-module isomorphic Hilbert extensions iff they are equivalent. The essential step of the solution is the application of the standard DR-theory to the assigned subcategory.