Odd-order abelian group not is fully invariant in holomorph

Statement
It is possible to have an fact about::odd-order abelian group $$G$$ such that $$G$$ is not a fact about::fully invariant subgroup in its holomorph.

The smallest counterexample is $$\Z_{p^2} \times \Z_p$$, i.e., the direct product of the cyclic groups of order $$p^2$$ and $$p$$.

Related facts

 * 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
 * Cyclic group not is characteristic in holomorph, cyclic group not is fully invariant in holomorph
 * Additive group of a field implies characteristic in holomorph
 * Odd-order elementary abelian group is fully invariant in holomorph