On the Fourier–Haar series of composite functions

The author determines the class of all homeomorphic changes of variable that preserve absolute convergence of the series of Fourier–Haar coefficients.
It is established that among all the continuously differentiable homeomorphic changes of variable only the functions $\varphi_1$ and $\varphi_2$ defined by the equalities $\varphi_1(x)=x$ and $\varphi_2(x)=1-x$ for $x\in [0,1]$ preserve both convergence and absolute convergence of the Fourier–Haar series.
The class of Borel measurable functions whose Fourier–Haar series converge everywhere under any homeomorphic change of variable is determined, along with the class of all Borel measurable functions whose Fourier–Haar series converge absolutely at every point under any homeomorphic change of variable.