  Let $d(n;\ell_1,M_1,\ell_2,M_2)$ denote the number of factorizations
$n=n_1n_2$, where each of the factors $n_i\in\mathbb{N}$ belongs to a
prescribed congruence class $\ell_i\bmod M_i\,(i=1,2)$. Let
$\Delta(x;\ell_1,M_1,\ell_2,M_2)$ be the error term of the asymptotic formula
of $\sum\limits_{n\leqslant x}d(n;\ell_1,M_1,\ell_2,M_2)$. In this paper, we
establish an asymptotic formula of the fourth power moment of
$\Delta(M_1M_2x;\ell_1,M_1,\ell_2,M_2)$ and prove that \begin{equation*}
  \int_1^T\Delta^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)\mathrm{d}x=\frac{1}{32\pi^4}C_4\Big(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\Big)
  T^2+O(T^{2-\vartheta_4+\varepsilon}), \end{equation*} with $\vartheta_4=1/8$,
which improves the previous value $\theta_4=3/28$ of K. Liu.
