Summary: For any graph \(G\), let \(G^\ast\) be the symmetric digraph obtained from \(G\) by replacing every edge with a pair of symmetric arcs. In this paper, we show that the necessary and sufficient condition for the existence of an \(\overline{S}_k\)-factorization in \((C_m\circ \overline{K}_n)^\ast\) is \(n\equiv 0\pmod {\frac{k(k-1)}{2}}\), where \(k>3\) is odd. In fact, our result deduces the result of Ushio on symmetric complete tripartite digraphs as a corollary. Further, a necessary condition and some sufficient conditions for the existence of an \(\overline{S}_k\)-factorization in \(K_{n_1,n_2,\ldots,n_m}^\ast\) are obtained.