  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.
