Every group retraction can be realized as a topological retraction

Given a group retraction $r: G \rightarrow H$, we construct a finite topological space $X_r$ of height 1, together with a topological retraction $\overline{r}: X_r \rightarrow X_r$, such that the group of automorphisms $\mathrm{Aut}(X_r)$ (or the group of self-homotopy equivalences $\mathcal{E}(X_r)$) of $X_r$ is isomorphic to $G$, and $\mathrm{Aut}(\overline{r}(X_r))$ (or $\mathcal{E}(\overline{r}(X_r))$) is isomorphic to $H$. Moreover, there is a natural map $\overline{r}' : \mathrm{Aut}(X_r) \rightarrow \mathrm{Aut}(\overline{r}(X_r))$ 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.