  The article deals with the problem of finding vertex-minimal graphs with a
given automorphism group. We exhibit two undirected $16$-vertex graphs having
automorphism groups $A_{4}$ and $A_{5}$. It improves the Babai's bound for
$A_{4}$ and the graphical regular representation bound for $A_{5}$. The graphs
are constructed using projectivisation of the vertex-face graph of icosahedron.
