Free group of Hamel bijections of big size

Abstract

A $f:\mathbb{R}\to \mathbb{R}$ is called Hamel function if its graph is a Hamel basis of the linear space ${\mathbb{R}}^2$ over rationals. We construct, assuming CH, a free group of the size $2^{\mathfrak{c}}$ contained in the class of all Hamel functions, with the indentity function included. This answers, consistently, a question posed recently by M. Lichman, M. Pawlikowski, S. Smolarek, and J. Swaczyna.

The result is false and it can't be corrected