  Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a
topological space $X_f$ such that its group of homeomorphisms is isomorphic to
$G$, its group of homotopy classes of self-homotopy equivalences is isomorphic
to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the
group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely
$f$. In addition, realization problems involving homology, homotopy groups and
groups of automorphisms are considered.
