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.