I am considering $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ with an automorphism that preserves an ample divisor class.

For an automorphism $\rho$ of a $K3$ surface, let ${\rm Pic}^\rho(S) = \{ D \in {\rm Pic}(S) \mid \rho^*(D) = D \}$.

Let me state the question:

Does there exist a triple $(H, S, \rho)$ of an ample divisor class $H$ of $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$, a smooth $K3$ surface $S$ that is an anti-canonical section of $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1 $ and an automorpism $\rho$ of $S$ such that $${\rm Pic}^\rho(S) = \left \langle H|_S \right \rangle?$$

According to Arnaud BEAUVILLE (https://arxiv.org/pdf/math/0211313), a $20-3=17$ dimensional family of $K3$ surfaces lie inside $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1 $. I do not see any obstacles which prevent such a triple from existing.

Perhaps, the Torelli theorem may help.

Does anyone have an answer?