We prove that the natural permutation representation of highly transitive finite groups cannot be realized as the full automorphism group of an idempotent, finite-dimensional evolution algebra acting on the set of lines spanned by its natural elements. Specifically, for any sufficiently large integer n and \(k \ge 4\) , there does not exist an idempotent evolution algebra X of dimension n such that \(\operatorname {Aut}(X)\) is isomorphic to a proper k-transitive subgroup of \(S_n\) . Nevertheless, we show that for any finite group G, any permutation representation \(\xi :G \rightarrow S_n\) , and any field \(\mathbb {k}\) , there exists an idempotent, finite-dimensional evolution \(\mathbb {k}\) -algebra X such that \(\operatorname {Aut}(X) \cong G\) , and the induced representation of \(\operatorname {Aut}(X)\) on the natural idempotents of X is equivalent to \(\xi \) .
