Converse of the internal-automorphism criterion for Frucht's theorem

Let $M$ be a model of set theory satisfying the Axiom of Choice, and let $M'=M/\mathcal{F}$ be a permutation model inside $M$. Suppose that Frucht's Theorem fails in $M'$. An internal automorphism is an automorphism belonging to the relevant model, and it is non-trivial when it is not the identity. Converse of the internal-automorphism criterion. Every $F\in\mathcal{F}$ contains a non-trivial internal automorphism. The conjecture proposes a converse to the stated criterion that non-trivial internal automorphisms force the failure of Frucht's Theorem; its status is not established in the supplied text.