  In this paper, we extend Ginzburg-Rallis' integral representation for the
exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that
of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its
analytic properties to determine the poles of this $L$-function. Furthermore,
we introduce the automorphic induction for automorphic representations of ${\rm
GU}_n$ and then show that the weak Langlands functorial lift for the
automorphic induction exists for generic cuspidal automorphic representations.
By using this automorphic induction, we give a conjectural criterion on the
existence of poles of $L(s,\pi,\wedge^{3}\otimes\chi)$ for discrete automorphic
representations in the tempered spectrum.
