Cyclic group not is characteristic in holomorph

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup not satisfying a particular subgroup property (namely, Characteristic subgroup (?)) in a particular group or type of group .

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