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.