G-birationally rigid cubic threefolds

We classify pairs $(X,G)$ consisting of a (possibly singular) cubic threefold $X\subset\mathbb{P}^4$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally rigid, i.e., $X$ is a $G$-Mori fiber space (over a point), and $X$ is not $G$-birational to any $G$-Mori fibre space that is not $G$-biregular to $X$.