Let G be a group and φ∈ Aut (G). Then the set G equipped with the binary operation a∗ b= φ(ab- 1) b gives a quandle structure on G, denoted by Alex (G, φ) , and called the generalised Alexander quandle of G with respect to φ. When G is an additive abelian group and φ= - id G, then Alex (G, φ) is the well-known Takasaki quandle of G. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if G≅ (Z/ pZ) n and φ is multiplication by a non-trivial unit of Z/ pZ, then Aut (Alex (G, φ)) acts doubly transitively on Alex (G, φ). This generalises a recent result of Ferman et al. (J Knot Theory Ramifications 20:463–468, 2011) for quandles of prime order.