Odd-order cyclic group is fully invariant in holomorph

Statement

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

Related facts

• Odd-order abelian group not is fully invariant in holomorph: The analogous statement is not true for odd-order abelian groups.
• Cyclic group not is fully invariant 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.
• Cyclic group not is characteristic in 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).