Primitive quandles with alternating displacement group

We classify primitive quandles with alternating displacement group. All of them are conjugation quandles, and the following is a complete list of the underlying conjugacy classes: transpositions in $S_n$ for $n=3$ and $n\geq5$; fixpoint-free involutions in $S_n$ for $n\equiv 2\pmod 4$, $n\geq6$; fixpoint-free involutions in $A_n$ for $n\equiv 0\pmod 4$, $n\geq12$; and one exceptional conjugacy class of size 36 in a group of order 720.