  In this paper, we study weighted $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$
a Muckenhoupt $A_{p}$ weight) of singular integrals with homogeneous
convolution kernel $K(x)$ on an arbitrary homogeneous group $\mathbb H$ of
dimension $\mathbb{Q}$, {under the assumption that $K_0$, the restriction of
$K$ to the unit annulus, is mean zero and $L^{q}$ integrable for some
$q_{0}<q\leq \infty$,} where $q_{0}$ is a fixed constant depending on $w$. We
obtain a quantitative weighted bound, which is consistent with the one obtained
by Hyt\"onen--Roncal--Tapiola in the Euclidean setting, for this operator on
$L^{p}(w)$. Comparing to the previous results in the Euclidean setting, our
assumptions on the kernel and on the underlying space are weaker. Moreover, we
investigate the quantitative weighted bound for the bi-parameter rough singular
integrals on product homogeneous Lie groups.
