Global liftings between inner forms of GSp(4)

For reductive groups G over a number field we discuss automorphic liftings of cohomological cuspidal irreducible automorphic representations π of $G\left(A\right)$ to irreducible cohomological automorphic representations of $H\left(A\right)$ for the quasi-split inner form H of G, and other inner forms as well. We show the existence of nontrivial weak global cohomological liftings in many cases, in particular for the case where G is anisotropic at the archimedean places. A priori, for these weak liftings we do not give a description of the precise nature of the corresponding local liftings at the ramified places, nor do we characterize the image of the lifting. For inner forms of the group $H=\mathrm{GSp}\left(4\right)$ however we address these finer questions. Especially, we prove the recent conjectures of Ibukiyama and Kitayama on paramodular newforms of square-free level.