A remark on the number of automorphisms of some algebraic structures

In these notes we look at the following question: given a category $\mathcal{C}$ of algebraic structure (e.g. the category of groups, monoids, partial groups, …) and a rational $r\in \mathbb{Q}$, does there exists an element $x\in \mathcal{C}$ such that the size of its automorphism group ${\text{Aut}}_{\mathcal{C}}(x)$ divided by the size of $x$ (whatever that would means) is equal to $r$ ? To our knowledge, this question was introduced by Tărnăuceanu in the category of groups. Here, we answer positively to this question in the categories of evolution algebras, graphs, monoids, partial groups and posets.