  Let $F$ be a number field, and $\BA=\BA_F$. In this paper, first, we provide
a family of global Arthur parameters confirming all parts of a general
conjecture on the relation between the structure of Fourier coefficients and
the structure of global Arthur parameters, given by Jiang in 2012. Then we
extend a correspondence between certain automorphic forms on $Sp_{4n}(\BA)$ and
$\wt{Sp}_{2n}(\BA)$, given by Ginzburg, Jiang and Soudry in 2012, to certain
automorphic forms on $Sp_{4mn}(\BA)$ and $\wt{Sp}_{4mn \pm 2n}(\BA)$, using the
same idea of considering compositions of automorphic descent maps.
