  In this article, we establish an asymptotic estimate on the number of
cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$
which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained
from symmetric cube transfer of automorphic representations of ${\rm
GL}_2(\mathbb A_{\mathbb Q})$ of a given weight and with varying level
structure. This generalises the recent work of C. Ambi [2020] about the similar
problem for ${\rm GL}_3$.
