Join of abelian subgroups of maximum order is intermediately characteristic

Statement
Let $$P$$ be a group of prime power order. Let $$J(P)$$ denote the join of abelian subgroups of maximum order in $$P$$, i.e., $$J(P)$$ is the subgroup generated by all abelian subgroups of maximum order in $$P$$. Then, $$J(P)$$ is an intermediately characteristic subgroup of $$P$$: for any subgroup $$Q$$ of $$P$$ containing $$J(P)$$, $$J(P)$$ is characteristic in $$Q$$.

In fact, $$J(P)$$ is the join of abelian subgroups of maximum order in $$Q$$.

Related facts

 * Join of abelian subgroups of maximum order not is image-closed characteristic
 * Join of abelian subgroups of maximum order in Sylow subgroup is characteristic in every p-subgroup containing it