A remark on the number of automorphisms of finite abelian groups

Let $A{b}_0$ be the class of finite abelian groups and consider the function $f:A{b}_0⟶(0,\mathrm{\infty })$ given by $f(G)=\frac{\mathrm{\mid }\mathrm{A}\mathrm{u}\mathrm{t}(G)\mathrm{\mid }}{\mathrm{\mid }G\mathrm{\mid }}$,, where $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ is the automorphism group of a finite abelian group $G$. In this short note, we prove that the image of $f$ is a dense set in $[0,\mathrm{\infty })$.