Odd-order cyclic group is fully invariant in holomorph

Statement

Any Odd-order cyclic group (?) is a Fully characteristic subgroup (?) inside its holomorph.

Related facts

• Odd-order Abelian group not is fully characteristic in holomorph: The analogous statement is not true for odd-order Abelian groups.
• Cyclic group not is fully characteristic in holomorph: The analogous statement is not true if we remove the conditions of odd order. In fact, if $4$ divides the order of a cyclic group, then it is not fully characteristic in its holomorph.

Facts used

1. Odd-order cyclic group equals commutator subgroup of holomorph
2. Commutator subgroup is fully characteristic

Proof

The proof follows directly from facts (1) and (2).