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.