p-groups with p² as a codegree

Let $G$ be a $p$-group and let $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is given by $|G:\text{ker}(\chi)|/\chi(1)$. This paper investigates the relationship between the nilpotence class of a group and the inclusion of $p^2$ as a codegree. If $G$ is a finite $p$-group with coclass $2$ and order at least $p^5$, or coclass $3$ and order at least $p^6$, then $G$ has $p^2$ as a codegree. With an additional hypothesis this result can be extended to $p$-groups with coclass $n\ge 3$ and order at least $p^{2n}$.