Groups with two extreme character degrees and their normal subgroups

We study the finite groups $G$ for which the set $\operatorname{cd}(G)$ of irreducible complex character degrees consists of the two most extreme possible values, that is, $1$ and $\vert G:Z(G)\vert^{1/2}$. We are easily reduced to finite $p$-groups, for which we derive the following group theoretical characterization: they are the $p$-groups such that $\vert G:Z(G)\vert$ is a square and whose only normal subgroups are those containing $G'$ or contained in $Z(G)$. By analogy, we also deal with $p$-groups such that $\vert G:Z(G)\vert=p^{2n+1}$ is not a square, and we prove that $\operatorname{cd}(G) =\{1,p^n\}$ if and only if a similar property holds: for any $N\trianglelefteq G$, either $G'\le N$ or $\vert NZ(G):Z(G)\vert\le p$. The proof of these results requires a detailed analysis of the structure of the $p$-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than $2$, then the index of the centre is small, and in some cases we may even bound the order of $G$.