Cyclic group not is characteristic in holomorph

Statement
A cyclic group need not be a characteristic subgroup inside its holomorph. In fact, if $$8$$ divides the order of the cyclic group, it is not characteristic inside its holomorph.

Related facts

 * Cyclic group not is fully invariant in holomorph: This breaks down even when $$4$$ divides the order of the group.
 * 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 elementary abelian group is fully invariant in holomorph
 * Odd-order abelian group not is fully invariant in holomorph