In this paper, we introduce the new construction of quandles. For a group G and a subset A of G we construct a quandle Q(G,A) which is called the (G,A)-quandle and study properties of this quandle. In particular, we prove that if Q is a quandle such that the natural map Q GQ from Q to the enveloping group GQ of Q is injective, then Q is the (G,A)-quandle for an appropriate group G and a subset A of G. Also we introduce the free product of quandles and study this construction for (G,A)-quandles. In addition, we classify all finite quandles with enveloping group a&2. ;copy 2020 World Scientific Publishing Company.