The setk $K_p$ in some finite groups

The study of the properties of the set $K_p$ consisting of elements of a non-Abelian group that commute with exactly $p$ elements of the group $G$ is continued. In particular, this question is considered for groups of order $p_1p_2\cdots p_k$, $k\geqslant 3$ and $p^2q$, where $p_i$, $q$ are prime numbers.
It is also proved that the set $K_5$ is non-empty in the three-dimensional projective special linear group. This group has the same order as the alternating group $A_8$, in which the set $K_5$ is empty.