Cyclic group not is characteristic in holomorph

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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

Proof