Existence of normal subgroups in finite p-groups

Suppose p is a prime, S is a finite p-group, and B is a subgroup of S of order pn and class at most c. Does S possess a normal subgroup of order pn and class at most c J. Alperin showed that the answer is negative in general. In this article, we show that the answer is affirmative if n is sufficiently small relative to p.