Holomorph of elementary abelian group not implies every automorphism is inner

Statement
It is possible to have an fact about::elementary abelian group $$G$$ such that the holomorph of $$G$$ is not a fact about::complete group. In fact, we can construct examples where the holomorph is not a fact about::group in which every automorphism is inner.

The smallest example is the particular example::elementary abelian group of order eight.

Opposite facts for abelian case

 * Holomorph of cyclic group of odd prime order is complete: This in turn can be deduced from the fact that semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner.

Analogue for non-abelian case

 * Characteristically simple and non-abelian implies automorphism group is complete

Other related facts about holomorphs

 * Additive group of a field implies monolith in holomorph, additive group of a field implies characteristic in holomorph
 * Odd-order elementary abelian group is fully invariant in holomorph
 * Odd-order cyclic group equals commutator subgroup of holomorph, odd-order cyclic group is fully invariant in holomorph, odd-order cyclic group is characteristic in holomorph
 * Odd-order abelian group not is fully invariant in holomorph