  In this paper, we give an asymptotic formula for the second moment of
Dirichlet twists of an automorphic $L$-function $L(s, \pi)$ on the critical
line averaged over characters and conductors, where $\pi$ denotes an
irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$
with unitary central character. We give some hybrid bound for the error term
with respect to the size of conductors of Dirichlet characters and that of the
automorphic representation.
