18.46 (2014)

Every finite group $G$ can be embedded in a group $H$ in such a way that every element of $G$ is a square of an element of $H$. The overgroup $H$ can be chosen such that $|H| \leqslant 2|G|^2$. Is this estimate sharp?

It is known that the best possible estimate cannot be better than $|H| \leqslant |G|^2$ (D. V. Baranov, Ant. A. Klyachko, Siber. Math. J., 53, no. 2 (2012), 201–206).