  We establish that the natural permutation representation of highly transitive
groups cannot be realized as the full group of automorphisms of an idempotent
finite-dimensional evolution algebra acting on the set of lines spanned by
natural elements. For instance, for any sufficiently large integer $n$, and
$k\geq 4$, there exists no idempotent evolution algebra $X$ such that $\dim
X=n$, and $\operatorname{Aut}(X)$ is isomorphic to a proper $k$-transitive
subgroup of $S_n$. However, we demonstrate that for any (not necessarily
faithful) permutation representation $\xi\colon G \to S_n$ and any field
$\Bbbk$, there exists an idempotent finite-dimensional evolution
$\Bbbk$-algebra $X$ such that $\operatorname{Aut}(X)\cong G$ and the induced
representation given by the $\operatorname{Aut}(X)$-action on the natural
idempotents of $X$ is equivalent to $\xi$.
