Given a group retraction r:G→H, we construct a finite topological space Xr of height 1, together with a topological retraction r:Xr→Xr
Nearby in the stack
, 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(r(Xr))
(or
E(r(Xr))
) is isomorphic to
H
. Moreover, there is a natural map
r′:Aut(Xr)→Aut(r(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.