In 2015, A.N.Skiba in [1] introduce definition: A subgroup H of G is said to be \({\sigma }\) -subnormal in G if there is a subgroup chain \(H=H_{0} \le H_{1} \le \cdots \le H_{t}=G\) such that either \(H_{i-1}\trianglelefteq H_{i}\) or \(H_{i}/(H_{i-1})_{H_{i}}\) is \(\sigma \) -primary for all \(i=1, \ldots , t\) . Later, Wenbin Guo and A.N.Skiba in [2] introduce the definition of \(\sigma \) -semipermutable: A subgroup H of G is said to be \(\sigma \) -semipermutable in G if G possesses a complete Hall \(\sigma \) -set \({\mathcal {H}}\) such that \(HA^{x}=A^{x}H\) for all \(A\in {{\mathcal {H}}}\) and all \(x\in G\) such that \(\sigma (A)\cap \sigma (H)= \emptyset \) . In this paper, we present a new generalized supplemented definition: A subgroup H of G is said to be: weakly \(\sigma \) -semipermutable in G if there exists a \(\sigma \) -subnormal subgroup T of G such that \(G=HT\) and \(H\cap T\le H_{\overline{\sigma } G}\) , where \(H_{\overline{\sigma } G}\) is the subgroup of H generated by all those subgroups of H which are \(\sigma \) -semipermutable in G. Also, the structure of a finite group with some weakly \(\sigma \) -semipermutable subgroups is investigated.
