Points in linear general position and automorphisms

Question

Let $S=\{p_1,\ldots,p_n\}\subset\mathbb{P}^2$ be a subset of points of the projective plane. Assume that there exists a subset $A\subseteq S$ of five points in linear general position (meaning that no three of these 5 points lie on the same line). Assume that there is an automorphism $\phi$ of $\mathbb{P}^2$ mapping $S$ to itself. May we conclude that then $\phi$ must be the identity?