Abelian-to-normal replacement theorem for prime exponent

Statement
Suppose $$P$$ is a finite fact about::group of prime exponent: group of prime power order, say $$p^r$$, and with exponent $$p$$ (so every element has order $$p$$). Suppose $$A$$ is an abelian subgroup of order $$p^n$$, and nilpotency class at most $$p + 1$$.

Then, there exists an fact about::abelian normal subgroup $$B$$ of $$P$$ such that:


 * $$B$$ is contained in the normal closure of $$A$$ in $$P$$
 * $$B$$ has the same order (i.e., $$p^n$$) as $$A$$

Related facts

 * Mann's replacement theorem for subgroups of prime exponent
 * p-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p
 * Group of exponent p and order greater than p^p is not embeddable in a maximal class group