  For reductive groups $G$ over a number field we discuss automorphic liftings
from cuspidal irreducible automorphic representations $\pi$ of $G(\mathbb{A})$
to cuspidal irreducible automorphic representations on $H(\mathbb{A})$ for the
quasi-split inner form $H$ of $G$. We show the existence of cohomological
nontrivial weak global liftings in many cases. A priori these weak liftings do
not give a description of the precise nature of the corresponding local
liftings at the ramified places and in particular do not characterize the image
of the lift. For inner forms of the group $H=\mathrm{GSp}(4)$ however we
address these finer details. Especially, we prove the recent conjectures of
Ibukiyama and Kitayama on paramodular newforms of squarefree level.
