In Section 9 of the paper 

I. Shimada: [An algorithm to compute automorphism groups of \(K3\) surfaces and an application to singular \(K3\) surfaces](http://dx.doi.org/10.1093/imrn/rnv006), *Int. Math. Res. Not.* **2015**, No. 22, 11961-12014 (2015) [ZBL1333.14034](https://zbmath.org/?q=an:1333.14034)</cite>

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


  [1]: http://www.math.sci.hiroshima-u.ac.jp/shimada/preprints/AlgoAutK3/AlgoAutK3.pdf