  In this article, we establish an asymptotic lower bound estimate on the
contribution of cuspidal automorphic representations of ${\rm GL}_4(\mathbb
A_{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}_4$ which are obtained
from automorphic tensor product of two automorphic representations of ${\rm
GL}_2(\mathbb A_{\mathbb Q})$ of given weights and with varying level
structure. In the end, we also prove that the symmetric cube of a
representation of ${\rm GL}_2$ and the automorphic tensor product of two
representations of ${\rm GL}_2$ can not be equal (up to a twist by a character
of ${\rm GL}_1$) to each other, under the suitable assumptions on the
representations being cuspidal and cohomological.
