  In \cite{MR3284482} and \cite{MR3658191}, the twisted standard
$\mathcal{L}$-function $\mathcal{L}(s,\pi,\chi,st)$ of a cuspidal
representation $ \pi$ of the exceptional group of type $G_2$ was shown to be
represented by a family of new-way Rankin-Selberg integrals. These integrals
connect the analytic behaviour of $\mathcal{L}(s,\pi,\chi,st)$ with that of a
family of degenerate Eisenstein series $\mathcal{E}_E(\chi, f_s, s, g)$ on
quasi-split forms $H_E$ of $Spin_8$, induced from Heisenberg parabolic
subgroups. The analytic behaviour of the series $\mathcal{E}_E(\chi, f_s, s,
g)$ in the right half-plane $Re(s)>0$ was studied in \cite{SegalEisen}. In this
paper we study the residual representations associated with
$\mathcal{E}_E(\chi, f_s, s, g)$.
