Let $S$ be a K3 surface and $\iota$ be anti-symplectic involution of $S$. Suppose that $g$ is a Kahler-Einstein metric on $S$. My question is;

Why $\iota$ is an isometry of $S_J$ with respect to $g$ for any complex structure $J$ of $S$ obtained by a hyperKahler rotation?

Is this statement true for any holomorphic action of $S$?

Let $S$ be a K3 surface and $\iota$ be anti-symplectic involution of $S$. Suppose that $g$ is a Kahler-Einstein metric on $S$. My question is;

Why $\iota$ is an isometry of $S$ with respect to $g$? Is this true for any holomorphic action of $S$?

Edit $\iota$ is called anti-symplectic if it acts on $\Omega^{2,0}$ as $-id$.