Odd-order elementary abelian group is fully invariant in holomorph

This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
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Statement

Let $p$ be an odd prime and consider the Elementary abelian group (?) $G=C_{p}\times C_{p}\times \dots \times C_{p}$ with the product taken $r$ times. Then, $G$ is a Fully invariant subgroup (?) inside its holomorph.

Facts used

1. Additive group of a field implies monolith in holomorph
2. Monolith is fully invariant in co-Hopfian group (and in particular, in a finite group).

Proof

The proof follows from facts (1) and (2), along with the observations that: (i) any elementary abelian group is the additive group of a field, so we can use (1), and (ii) the holomorph of a finite group is again finite, so we can use (2).