Let $G$ be a group with the property $G=G_e\dot{\cup} G_O$ with $G_oG_o^{-1}\subseteq G_o^{-1}G_o=G_e\leq G$, and with no any element of order $2$ .

($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G_o^{-1}=\{x^{-1}: x\in G_O\}$)

We observe that $|G:G_e|=2$, $|G|=2|G_o|=2|G_e|$, and the integer numbers group, $S_n$ and $\mathbb{Z}_{2m}$ enjoy the property.

Now, how are these groups characterized?. Any ideas?, references?, classes of such groups?