R-equivalence of quandle colorings and inner automorphisms

R-equivalence is an equivalence relation on the set of colorings of an oriented knot diagram by a quandle. In this paper, we show that a certain subgroup of the inner automorphism group of a quandle acts on the R-equivalence class of a given coloring by the quandle. We also determine the R-equivalence classes of colorings of a diagram of a $2$-bridge knot by a dihedral quandle completely, under a certain condition.