Let $G$ be 'ax+b' topological group i.e subgroup of $GL(2,\mathbb R)$ containing $2\times 2 $ matrices of type $\{\left(\begin{matrix} a&b\\\0&1\end{matrix}\right): a\neq 0\; a,b\in \mathbb R\}$. Further let $A$ be an element of order $2$ in G. Is is true that only continuous functions /homeomorphisms $f$ on $G$ satisfying $f(AX)=f(X)A$ for all $X\in G$ are $f(x)=Bx^{1}$, where $B$ is any element of $G$.
