On some topological realizations of groups and homomorphisms

Given a homomorphism of groups f:G\rightarrow H, we construct a topological space X_f such that its group of homeomorphisms Aut(X_f) is isomorphic to G, its group of homotopy classes of self-homotopy equivalences \mathcal {E}(X_f) is isomorphic to H and the natural map between Aut(X_f) and \mathcal {E}(X_f) is f. In addition, we consider realization problems involving homology groups, homotopy groups and groups of automorphisms.