The minimal number of characters over a normal p-subgroup

If N is a normal p-subgroup of a finite group G and θ∈Irr(N) is a G-invariant irreducible character of N, then the number |Irr(G|θ)| of irreducible characters of G over θ is always greater than or equal to the number kp′(G/N) of conjugacy classes of G/N consisting of p′-elements. In this paper, we investigate when there is equality.