This paper focuses on ovoids of the Klein quadric \(Q^+ (5,q)\) admitting the symmetric group \(S_5\) as a group of automorphisms, and such that \(q\) and 30 are relatively prime, and \(q^2 - 1\) is not divisible by 5. Such an ovoid defines a translation plane of order \(q^2\) by the Klein correspondence. These ovoids are constructed by paisting together certain orbits. There are examples of infinite series, but the authors also give a lot of explicit examples for the cases \(q = 7,13,17\). They also prove some results on the automorphism groups of such ovoids. In particular, it is shown that the translation complement of the translation planes corresponding to the infinite series of examples, is non-solvable and has an order which is coprime to the characteristic.