  Frucht showed that, for any finite group $G$, there exists a cubic graph such
that its automorphism group is isomorphic to $G$. For groups generated by two
elements we simplify his construction to a graph with fewer nodes. In the
general case, we address an oversight in Frucht's construction. We prove the
existence of cycle double covers of the resulting graphs, leading to simplicial
surfaces with given automorphism group. For almost all finite non-abelian
simple groups we give alternative constructions based on graphic regular
representations. In the general cases $C_n,D_n,A_5$ for $n\geq 4$, we provide
alternative constructions of simplicial spheres. Furthermore, we embed these
surfaces into the Euclidean 3-Space with equilateral triangles such that the
automorphism group of the surface and the symmetry group of the corresponding
polyhedron in $\mathrm{O}(3)$ are isomorphic.
