The Reidemeister spectrum of split metacyclic groups

Given a group $G$ and an automorphism $\phi$ of $G$, two elements $x, y \in G$aresaidtobe$\varphi$-conjugateif$x = gy \varphi(g)^{-1}$forsome$g \in G$. The number of equivalence classes for this relation is the Reidemeister number $R(\phi )$ of $\phi$. The set ${R(\psi) \mid \psi \in $\mathrm{A}\mathrm{u}\mathrm{t}$(G)}$iscalledtheReidemeisterspectrumof$G$. We fully determine the Reidemeister spectrum of split metacyclic groups of the form $C_n⋊C_p$ where $p$ is a prime and the action is non-trivial.