Automorphisms of some magmas of order $k+k^2$

This paper is devoted to the study of automorphisms of finite magmas and to the representation of the symmetric permutation group $ S_k $ and some of its subgroups by automorphism groups of finite magmas. The theory that studies automorphism groups of magmas is well developed and is represented by a multitude of works, when magma is a quasigroup, semigroup, loop, monoid or group. There are also studies in which problems related to the study of automorphisms of magmas that are not a semigroup or quasigroup are considered.
In this paper, we introduce some finite magmas $ \mathfrak{S} = (V, *) $ of order $ k + k^2 $. For magma $\mathfrak{S}$ it was possible to describe the automorphism group and write down the general form of the automorphism. In addition, the connection between automorphisms of magmas $\mathfrak{S}$ and permutations of a finite set of $ k $ elements has been revealed. All automorphisms of magma $\mathfrak{S}$ are parametrized by permutations from a certain subgroup (a description of this subgroup is given) of the symmetric permutation group $ S_k $.
In addition, it is established that the group $ S_k $ is isomorphic to the group of all automorphisms $ Aut \ (\mathfrak{S}) $ of a suitable magma $ \mathfrak{S}$ of order $ k + k ^ 2 $.