Splittability, non-splittability, and automatic splittability of metacyclic $p$-groups

Let $P$ be a finite metacyclic $p$-group where $p$ is an odd prime. We refine the notions split metacyclic and non-split metacyclic group by distinguishing whether a metacyclic structure (i.e. a cyclic normal subgroup $K$ of $P$ such that $P\mathrm{/}K$ is also cyclic) is non-splittable or splittable or automatically split. We show that these different types can be characterised in terms of $\mathrm{\mid }K\mathrm{\mid }$. We also determine which of these types can coexist on the same group $P$.