  In this paper we classify three-dimensional singular cubic hypersurfaces with
an action of a finite group $G$, which are not $G$-rational, are not
$G$-birationally isomorphic to a quadric and have no birational structure of
$G$-Mori fiber space with the base of positive dimension. Also we prove the
$A_{5}$ -birational superrigidity of the Segre cubic.
