Endomorphisms of some groupoids of order $k+k^2$

Automorphisms and endomorphisms are actively used in various theoretical studies. In particular, the theoretical interest in the study of automorphisms is due to the possibility of representing elements of a group by automorphisms of a certain algebraic system. For example, in 1946, G. Birkhoff showed that each group is the group of all automorphisms of a certain algebra. In 1958, D. Groot published a work in which it was established that every group is a group of all automorphisms of a certain ring. It was established by M. M. Glukhov and G. V. Timofeenko: every finite group is isomorphic to the automorphism group of a suitable finitely defined quasigroup.
In this paper, we study endomorphisms of certain finite groupoids with a generating set of $k$ elements and order $k + k^2$, which are not quasigroups and semigroups for $k>1$. A description is given of all endomorphisms of these groupoids as mappings of the support, and some structural properties of the monoid of all endomorphisms are established. It was previously established that every finite group embeds isomorphically into the group of all automorphisms of a certain suitable groupoid of order $ k + k^2$ and a generating set of $k$ elements.
It is shown that for any finite monoid $G$ and any positive integer $k\ge|G| $ there will be a groupoid $S$ with a generating set of $k$ elements and order $k+k^2$ such that $G$ is isomorphic to some submonoid of the monoid of all endomorphisms of the groupoid $S$.