On the finite near-rings generated by endomorphisms of an extra-special 2-group

We consider the near-rings generated by endomorphisms of some extra-special 2-groups. The most essential difference of a near-ring from a usual ring is the absence of the second distributivity. In this paper, we prove that the near-ring $E(G)$ generated by endomorphisms of an extra-special 2-group $G$ of order $2^{2n+1}$ has the order which divides $2^{2^{2n}+4n^2}$ and that the near-ring $E(G)$ of the extra-special 2-group $G$ of type $-$ of order $2^{2n+1}$ has the order divided by $2^{2^{2n}+4n^2-2}$. In this case, for $n=1$ and $n=2$ the upper bound is attainable: the near-ring $E(G)$ of the group $D_8$ has the order $2^8$, and the near-ring $E(G)$ of an extra-special 2-group $D_8\ast Q_8$ has the order $2^{32}$.