On weakly s-permutable subgroups of finite groups

Let G be a finite group, H a subgroup of G and HsG the subgroup of H generated by all those subgroups of H which are s-permutable in G. Then we say that H is weakly s-permutable in G if G has a subnormal subgroup T such that HT=G and T∩H⩽HsG. We fix in every non-cyclic Sylow subgroup P of G a subgroup D satisfying 1<|D|<|P| and study the structure of G under the assumption that all subgroups H with |H|=|D| are weakly s-permutable in G.