Free group of Hamel bijections of big size

A $f\colon\mathbb{R}\to\mathbb{R}$ 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.