Elementary abelian-to-2-subnormal replacement theorem

Hands-on statement
Suppose $$P$$ is a fact about::group of prime power order, say $$p^r$$, where $$p \ge 5$$. Let $$A$$ be an elementary abelian subgroup of $$P$$.

Then, there exists an elementary abelian subgroup $$A^*$$ of $$P$$ such that:


 * $$A^*$$ has the same order as $$A$$
 * $$A^*$$ is a 2-subnormal subgroup of $$P$$: it is normal in its normal closure in $$P$$.

Statement in terms of weak 2-subnormal replacement condition
For $$p$$ at least $$5$$ and any nonnegative integer $$k$$, the singleton set comprising the elementary abelian subgroup of order $$p^k$$ is a collection of subgroups satisfying a weak 2-subnormal replacement condition.