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

If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi (1)\le \mathrm{\mid }G\mathrm{/}Z(G){\mathrm{\mid }}^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fernández-Alcober and Moretó obtained the relation between the character degree set of a finite $p$-group $G$ and its normal subgroups depending on whether $\mathrm{\mid }G\mathrm{/}Z(G)\mathrm{\mid }$ is a square or not. In this paper we investigate the finite $p$-group $G$ where for any normal subgroup $N$ of $G$ with $G^{\mathrm{\prime }}\le ̸N$ either $N\le Z(G)$ or $\mathrm{\mid }NZ(G)\mathrm{/}Z(G)\mathrm{\mid }\le p$ and obtain some alternate characterizations of such groups. We find that if $G$ is a finite $p$-group with $\mathrm{\mid }G\mathrm{/}Z(G)\mathrm{\mid }=p^{2n+1}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G^{\mathrm{\prime }}\le ̸N$ or $N\le Z(G)$, then $cd(G)=\{1,p^n\}$. We also find that if $G$ is a finite $p$-group with nilpotency class not equal to $3$ and $\mathrm{\mid }G\mathrm{/}Z(G)\mathrm{\mid }=p^{2n}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G^{\mathrm{\prime }}\le ̸N$ or $\mathrm{\mid }NZ(G)\mathrm{/}Z(G)\mathrm{\mid }\le p$, then $cd(G)\subseteq \{1,p^{n-1},p^n\}$.