Automorphism groups of quandles and related groups

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{j}(G)$ of a group $G$ we determine several subgroups of $\mathrm{A}\mathrm{u}\mathrm{t}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group $\mathrm{A}\mathrm{u}\mathrm{t}(Q)$. In particular, we prove that ${\rm Aut}({\rm Conj}(G))={\rm Z}(G)\rtimes {\rm Aut}(G)$ifandonlyifeither${\rm Z}(G)=1$or$G$isoneofthegroups$$\mathbb{Z}$_2$,$$\mathbb{Z}$_2^2$or$$\mathbb{Z}$_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove that the group of inner automorphisms $\mathrm{I}\mathrm{n}\mathrm{n}(T(G))$ is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles $Q$ with $3\le k$-transitive action of $\mathrm{A}\mathrm{u}\mathrm{t}(Q)$.