The fraction of the bijections generating the near-ring of 0-preserving functions

Let $\langle G, + \rangle$ be a finite (not necessarily abelian) group. Then $M_0(G):=\{f:G \to G|f(0)=0\}$ is a near-ring, i.e., a group which is also closed under composition of functions. In Theorem 4.1 we give lower and upper bounds for the fraction of the bijections which generate the near-ring $M_0(G)$. From these bounds we conclude the following: If $G$ has few involutions and the order of $G$ is large, then a high fraction of the bijections generate the near-ring $M_0(G)$. Also the converse holds: If a high fraction of the bijections generate $M_0(G)$, then $G$ has few involutions (compared to the order of $G$).