Prime-base logarithm of order of 2-subnormal subgroup of group of prime power order gives upper bound on nilpotency class of its normal closure

Statement
Suppose $$P$$ is a group of prime power order (for the prime number $$p$$) and $$H$$ is a fact about::2-subnormal subgroup (in particular, a fact about::2-subnormal subgroup of group of prime power order). Suppose $$K$$ is the fact about::normal closure of $$H$$ in $$P$$. In particular, this means that $$H$$ is a normal subgroup of $$K$$ and $$K$$ is a normal subgroup of $$P$$.

Then, if the order of $$H$$ is $$p^r$$ (so that the fact about::prime-base logarithm of order is $$r$$) and the fact about::nilpotency class of $$K$$ is $$c$$, then $$c \le r$$.

Applications

 * Elementary abelian-to-normal replacement theorem for large primes