On the notion of projective isomorphism

Question

Assume that $X,Y$ are closed subschemes of some projective space $P$. I call $X,Y$ projectively isomorphic if there exists an isomorphism $f:X\rightarrow Y$ with $f^\ast O_Y(1)=O_X(1)$. Here $O_X(1)$ is the pullback/restriction of $O_P(1)$. Does this imply that $f$ comes from an automorphism of $P$? I can easily prove the converse.

Answer 1

I can take $X,Y$ to be 4 points in $\mathbb{P}^1$, e.g. $X=\{0,1,\infty,a\}$ and $Y=\{0,1,\infty,b\}$, with $a\ne b$. Then clearly there is an isomorphism $f:X\rightarrow Y$ with $f^*\mathcal{O}_Y(1) = \mathcal{O}_X(1)$ but this does not come from an automorphism of $\mathbb{P}^1$, since any such automorphism is determined by 3 points.