Inner twists for Siegel modular forms and the images of associated Galois representations

We study the group of inner twists for Siegel modular forms of genus two. For lifted Siegel modular forms (these are Siegel modular forms obtained from elliptic modular forms by various functorial lifts), we compute the group of inner twists in terms of inner twists of elliptic modular forms. On the other hand, for non-lifted Siegel modular forms, we provide necessary and sufficient conditions for the existence of inner twists in terms of the Lie algebras of the images of Galois representations.