Finite and automorphism group is transitive on non-identity elements implies elementary Abelian

Statement

Suppose $G$ is a finite group whose automorphism group acts transitively on the non-identity elements of $G$. In other words, $G$ is a Group whose automorphism group is transitive on non-identity elements (?). Then, $G$ is either trivial or an elementary Abelian group: it is a direct product of cyclic groups of order $p$ for some prime $p$.

Related facts

Breakdown for infinite groups

There exist infinite groups whose automorphism group is transitive on the non-identity elements, that are not even Abelian.

Other related facts

• Equivalence of definitions of additive group of a field: For an Abelian group, the automorphism group being transitive on non-identity elements is equivalent to the group being the additive group of some field.