Ibukiyama correspondences on automorphic forms on Mp4(AQ) and SO5(AQ) generating large discrete series representations at the real place

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 ${\mathrm{Mp}}_4\left(A_Q\right)$ or ${\mathrm{SO}}_5\left(A\right)$ 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.