On the set $K_{p}$ in finite groups

The properties of the set $K_{p}$ consisting of elements of a non-Abelian group commuting with exactly $p$ elements of the group are considered. In particular, the properties of the set $K_{p}$ in permutation groups and some solvable groups. One more proof is given that all involutions of a finite simple non-Abelian group $G$ with a nonempty set $K_{3}$ form one conjugacy class.