Extensions and the induced characters of cyclic p-groups

For a prime p, we denote by B-n the cyclic group of order p(n). Let phi be a faithful irreducible character of B-n, where p is an odd prime. We study the p-group G containing B-n such that the induced character phi(G) is also irreducible. The purpose of this article is to determine the subgroup N-G (N-G (B-n)) of G under the hypothesis [N-G (B-n) : B-n](4) <= p(n).