The Reidemeister spectrum of finite abelian groups

For a finite abelian group $A$, the Reidemeister number of an endomorphism $\phi$ equals the size of $\mathrm{F}\mathrm{i}\mathrm{x}(\phi )$, the set of fixed points of $\phi$. Consequently, the Reidemeister spectrum of $A$ is a subset of the set of divisors of $\mathrm{\mid }A\mathrm{\mid }$. We fully determine the Reidemeister spectrum of $\mathrm{\mid }A\mathrm{\mid }$, that is, which divisors of $\mathrm{\mid }A\mathrm{\mid }$ occur as the Reidemeister number of an automorphism. To do so, we discuss and prove a more general result providing upper and lower bounds on the number of fixed points of automorphisms related to a given automorphism $\phi$.