The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio $$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If $P(G)$ is the monoid of subsets of $G$ with respect to the usual product of group subsets, then the commutativity degree $d(P(G))$ of $P(G)$ can be defined similarly: $$d(P(G))=\frac{|\{(A,B)\in P(G)^2 | AB=BA\}|}{|P(G)|^2}.$$Which are the connections between $d(G)$ and $d(P(G))$? Is there a constant $c\in (0,1)$ such that $d(P(G))\leq c$ for any finite non-abelian group $G$?