Automorphisms of quartic surfaces and Cremona transformations

In this paper, we consider the problem of determining which automorphisms of a smooth quartic surface $S\subset P^3$ are induced by a Cremona transformation of $P^3$. We provide the first steps towards a complete solution of this problem when $\rho \left(S\right)=2$. In particular, we give several examples of quartics whose automorphism groups are generated by involutions, but no non-trivial automorphism is induced by a Cremona transformation of $P^3$, giving a negative answer for Oguiso's question of whether every automorphism of finite order of a smooth quartic surface $S\subset P^3$ is induced by a Cremona transformation.