Every group retraction can be realized as a topological retraction

Given a group retraction r: G → H, we construct a finite topological space Xᵣ of height 1, together with a topological retraction r̄: Xᵣ → Xᵣ, such that the group of automorphisms Aut(Xᵣ) (or the group of self-homotopy equivalences E(Xᵣ)) of Xᵣ is isomorphic to G, and Aut(r̄(Xᵣ)) (or E(r̄(Xᵣ))) is isomorphic to H. Moreover, there is a natural map r̄' : Aut(Xᵣ) → Aut(r̄(Xᵣ)) that coincides with the original group retraction r. As a direct consequence of this construction, we show that height 1 is the minimal height required to realize any finite group as the group of automorphisms (or the group of self-homotopy equivalences) of a finite topological space, except in the case where G is a symmetric group. In that unique case, the group can be realized by a finite topological space of height 0.