p-groups with small number of character degrees and their normal subgroups

If G be a finite p-group and χ is a non-linear irreducible character of G, then χ(1)≤ |G/Z(G)|1/2. In {fernandez2001groups}, Fern\'{a}ndez-Alcober and Moret\'{o} obtained the relation between the character degree set of a finite p-group G and its normal subgroups depending on whether $|G/Z(G)|$ is a square or not. In this paper we investigate the finite p-group G where for any normal subgroup N of G with G' ≤ N either N≤ Z(G) or |NZ(G)/Z(G)|≤ p and obtain some alternate characterizations of such groups. We find that if G is a finite p-group with |G/Z(G)|=p²ⁿ⁺¹ and G satisfies the condition that for any normal subgroup N of G either G' ≤ N or N≤ Z(G), then cd(G)=\1, pⁿ\. We also find that if G is a finite p-group with nilpotency class not equal to $3$ and |G/Z(G)|=p²ⁿ and G satisfies the condition that for any normal subgroup N of G either G' ≤ N or |NZ(G)/Z(G)|≤ p, then cd(G) ⊆ \1, pⁿ⁻¹, pⁿ\.