Bounds on squares of two-sets

Let G be a finite group and let pi(G) denote the proportion of (x, y) ∈ G2 for which the set {x2,xy,yx,y2} has cardinality i. We show that either 0 < pi(G) + p2(G) ≤ 1/2 or pi(G) + p2(G) = 1, and that either p4(G) = 0 or 5/32 ≤ p4(G) ≤ 1. Each of the preceding inequalities are the best possible.