Automorphism groups of quandles and related groups

In this paper we study various questions concerning automorphisms of quandles. For a conjugation quandle (Formula presented.) of a group G we determine several subgroups of (Formula presented.) and find necessary and sufficient conditions for these subgroups to coincide with the whole group (Formula presented.). In particular, we prove that (Formula presented.) if and only if either (Formula presented.) or G is one of the groups (Formula presented.), (Formula presented.) or (Formula presented.). 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 (Formula presented.) 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 k-transitive action of (Formula presented.) for (Formula presented.).