There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$


Let $\mathbb{Z}/2=\{ 1,s \}$ be the group with 2 elements. 


My question is: is there an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ such that it is compatible with the $SL(2,\mathbb{R})$ action and the action of the nontrivial element $s$ has no fixed point.


By compatible with the $SL(2,\mathbb{R})$ action I mean $\forall g\in SL(2,\mathbb{R}), x\in \mathbb{C}P^1$, we have
$$
g\cdot(s\cdot x)=s\cdot(g\cdot x).
$$


The action of $s$ need to be a diffeomorphism of $\mathbb{C}P^1=S^2$ but not required to be holomorphic.



Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$  which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$
Such $s$ does not exist since it would have to fix fixed points of all parabolic elements of $SL(2,R)$ and, hence, a circle. Note that your compatibility notion is usually called commutation. 


