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 fact about::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 $$p$$.

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.