Involutions with given weight deficit corresponding to the Cayley table of the finite Abelian group

We investigate the weight characteristics of involutions over finite Abelian groups $G_n$ of order $n\geqslant3$. For random equiprobable involution the distribution of the number of its binary cycles coinciding with binary cycles of fixed involution is found, the convergence of this distribution to the Poisson distribution with the parameter $\lambda=\frac12$ as $n\to\infty$ is proved. Mean value of the deficit of random equiprobable convolution is computed.