  In this paper, we study the $L^{p}$ boundedness and $L^{p}(w)$ boundedness
($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of fractional maximal
singular integral operators $T_{\Omega,\alpha}^{\#}$ with homogeneous
convolution kernel $\Omega(x)$ on an arbitrary homogeneous group $\mathbb H$ of
dimension $\mathbb{Q}$. We show that if $0<\alpha<\mathbb{Q}$, $\Omega\in
L^{1}(\Sigma)$ and satisfies the cancellation condition of order $[\alpha]$,
then for any $1<p<\infty$, \begin{align*}
\|T_{\Omega,\alpha}^{\#}f\|_{L^{p}(\mathbb{H})}\lesssim\|\Omega\|_{L^{1}(\Sigma)}\|f\|_{L_{\alpha}^{p}(\mathbb{H})},
\end{align*} where for the case $\alpha=0$, the $L^p$ boundedness of rough
singular integral operator and its maximal operator were studied by Tao
(\cite{Tao}) and Sato (\cite{sato}), respectively.
  We also obtain a quantitative weighted bound for these operators. To be
specific, if $0\leq\alpha<\mathbb{Q}$ and $\Omega$ satisfies the same
cancellation condition but a stronger condition that $\Omega\in L^{q}(\Sigma)$
for some $q>\mathbb{Q}/\alpha$, then for any $1<p<\infty$ and $w\in A_{p}$,
\begin{align*}
\|T_{\Omega,\alpha}^{\#}f\|_{L^{p}(w)}\lesssim\|\Omega\|_{L^{q}(\Sigma)}\{w\}_{A_p}(w)_{A_p}\|f\|_{L_{\alpha}^{p}(w)},\
\ 1<p<\infty. \end{align*}
