In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={ \rm Conj}(G)$ of a group $G$ we determine several subgroups of ${ \rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${ \rm Aut}(Q)$. In particular, we prove that ${ \rm Aut}({ \rm Conj}(G))={ \rm Z}(G)\rtimes { \rm Aut}(G)$ if and only if either ${ \rm Z}(G)=1$ or $G$ is one of the groups $\mathbb{ Z}_2$, $\mathbb{ Z}_2^2$ 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 ${ \rm Inn}(T(G))$ 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 $3\leq k$-transitive action of ${ \rm Aut}(Q)$.
Categories: math.GR math.GT