Let G be a topological group and a be an element of order 2 in G. Further suppose the element a does not belong to center of G. Then is it true that only homeomorphism f of G such that $f(ax)=f(x)a$ for all x in G is $f(x)=x^{1}$.

The answer is no. Take $H$ any topological group, $H'$ another topological group having a noncentral element $a'$ of order 2, $G=H\times H'$, $a=1_H\times a'$ and $f=h\times inv_{H'}$ where $h$ is any antiautomorphism of $H$ and $inv_{h'}$ is the map $h'\mapsto h'^{1},h'\in H'$. 

