  In our previous paper we gave proofs of Ibukiyama's correspondences on
holomorphic Siegel modular forms of degree 2 of half-integral weight and
integral weight. In this paper, we formulate and prove similar correspondences
on automorphic forms on $\operatorname{Mp}_4(\mathbb{A}_\mathbb{Q})$ or
$\operatorname{SO}_5(\mathbb{A}_\mathbb{Q})$ generating large discrete series
representations at the real components. In addition, we show that the
correspondences can be described in terms of local theta correspondences.
