In this note we discuss the effect of the B Z / p \textrm {B}\mathbb {Z}/p -nullification P B Z / p \mathbf {P}_{\textrm {B}\mathbb {Z}/p} and the B Z / p \textrm {B}\mathbb {Z}/p -cellularization C W B Z / p \mathbf {CW}_{\textrm {B}\mathbb {Z}/p} over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe P B Z / p B G \mathbf {P}_{\textrm {B}\mathbb {Z}/p}\textrm {B}G by means of a covering fibration, and we classify all finite groups G G for which B G \textrm {B}G is B Z / p \textrm {B}\mathbb {Z}/p -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of P B Z / p B G \mathbf {P}_{\textrm {B}\mathbb {Z}/p}\textrm {B}G and C W B Z / p B G . \mathbf {CW}_{\textrm {B}\mathbb {Z}/p}\textrm {B}G.