The oriented diffeomorphism group \(\mathcal{D}(L)\) of an \(n\)-component link \(L\) in \(S^3\) consists of all orientation-preserving diffeomorphisms of \(S^3\) that preserve the link setwise (also called the intrinsic symmetry group; the symmetry groups instead are diffeomorphisms modulo isotopy). The action of \(\mathcal{D}(L)\) on the components of \(L\) defines a homomorphism from \(\mathcal{D}(L)\) to the symmetric group \(\mathbb S_n\); denoting its image in \(\mathbb S_n\) by \(\mathbb S(L)\), the basic question considered in the paper is the following: which subgroups of \(\mathbb S_n\) arise as \(\mathbb S(L)\) for some \(n\)-component link \(L\)? (For example, \(\mathbb S(L) = \mathbb S_n\) for the trivial link \(L\) of \(n\) unknotted, unlinked components.) The main result of the present paper states that, for the alternating group \(\mathbb A_n\) with \(n \ge 6\), there does not exist an \(n\)-component link \(L\) with \(\mathbb S(L) =\mathbb A_n\). However there is an example of a 4-component link \(L\) with \(\mathbb S(L) = \mathbb A_4\) (found by Dunfield by computer using SnapPy), and it remains open whether there is a 5-component link \(L\) with \(\mathbb S(L) =\mathbb A_5\). The author discusses also an “orientation-reversing version”, replacing \(\mathbb S_n\) by the group \(\mathbb Z_2 \oplus ((\mathbb Z_2)^n \rtimes\mathbb S_n)\).