Finite groups with mostly involuted cyclic subgroups

Let $G$ be a finite group, define $I(G)=\{x\in G:x^2=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=\mathrm{\mid }I(G)\mathrm{\mid }$ and $c(G)=\mathrm{\mid }C(G)\mathrm{\mid }$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also prove that the range of the function given by $\beta (G)=\frac{i(G)}{c(G)}$ is dense in $[0,1]$.