K3 surfaces with small Picard number and symmetry

Question

I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution. Here, low is no mathematically precise concept. I want to do computations with Monad bundles and for that lower is better for me. For Picard rank bigger than 5 those computations become too difficult for me.

For example, the very general branched double cover of $\mathbb{CP}^2$ branched over a sextic has $Pic \simeq \mathbb{Z}$ and the map that swaps the two sheets of the cover is a holomorphic involution.

The post Picard groups of quartic K3 surfaces contains more examples of K3 surfaces with Picard ranks 1, 2, and 3, but I didn't find any holomorphic involutions of the K3 surfaces mentioned there.

Answer 1

In Section 9 of the paper

I. Shimada: An algorithm to compute automorphism groups of (K3) surfaces and an application to singular (K3) surfaces, Int. Math. Res. Not. 2015, No. 22, 11961-12014 (2015) ZBL1333.14034

there are many examples of complex elliptic K3 surfaces $X$ with Picard rank 3 and having (infinite) automorphism group containing involutions (in fact, $\operatorname{Aut}(X)$ contains a copy of $\mathbb{Z}/2 \ast \mathbb{Z}/2$).