Constructions of quandles over groups and rings

We present the following constructions: extension of a trivial quandle by a group with a nontrivial abelian subgroup, a quandle over a noncommutative group by an arbitrary antiautomorphism, a quandle presenting from a generalized Alexander quandle with the replacement of the automorphism by a central antiautomorphism, and a quandle over an $n$-dimensional module depending on $n(n-1)-1$ parameters with $n\geq 2$ in a commutative unital ring.