We are interested in Gizatullin’s problem which consists of the following question: Given a smooth quartic surface S ⊂ P3, which automorphisms of S are induced by Cremona transformations of P3? Cremona transformations of P3 can be written as a composition of a finite sequence of elementary maps. This is an algorithmic process called the Sarkisov Program. In this talk, we will solve Gizatullin’s problem when S ⊂ P3 has Picard number two by using the Sarkisov program. The results that will be presented are in collaboration with Ana Quedo, and with Carolina Araujo and Sokratis Zikas.