Let $P=\{p_1,\ldots,p_6\}\subset\mathbb{P}^1$ be a set of six general points of the projective line. In particular there are no two different subsets $\{p_{i_1},\ldots,p_{i_4}\}$ and $\{p_{j_1},\ldots,p_{j_4}\}$ of four elements of $P$ such that the $p_{i_k}$'s and the $p_{j_k}$'s have the same cross-ratio.

Assume that there exists a projective transformation $f:\mathbb{P}^1\rightarrow\mathbb{P}^1$ such that $f(P) = P$; that is $f$ acts on $P$ by permuting the $p_i$'s.

Do we have then that $f = \operatorname{Id}_{\mathbb{P}^1}$ ?