The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups

Let $G$ be a group. A subset $S$ of $G$ is said to normally generate $G$ if $G$ is the normal closure of $S$ in $G.$ In this case, any element of $G$ can be written as a product of conjugates of elements of $S$ and their inverses. If $g\in G$ and $S$ is a normally generating subset of $G,$ then we write $\Vert g{\Vert }_S$ for the length of a shortest word in ${\text{Conj}}_G(S^{\pm 1}):=\{h^{-1}sh|h\in G,s\in S\text{or}s{}^{-1}\in S\}$ needed to express $g.$ For any normally generating subset $S$ of $G,$ we write $\Vert G{\Vert }_S=\text{sup}\{\Vert g{\Vert }_S|g\in G\}.$ Moreover, we write $\mathrm{\Delta }(G)$ for the supremum of all $\Vert G{\Vert }_S,$ where $S$ is a finite normally generating subset of $G,$ and we call $\mathrm{\Delta }(G)$ the conjugacy diameter of $G.$ In this paper, we derive the conjugacy diameters of the semidihedral $2$-groups, the generalized quaternion groups and the modular $p$-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.