Infinite transitivity and special automorphisms

It is known that if the special automorphism group $\text{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we address the question whether this is the only possibility for the automorphism group $\text{Aut}(X)$ to act infinitely transitively on $X$. We show that this is the case provided $X$ admits a nontrivial $\mathbb{G}_a$- or $\mathbb{G}_m$-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.