On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups

Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP = PA for all Sylow subgroups P of G. The symbol HsG denotes the subgroup generated by all those subgroups of H which are S-quasinormal in G. A subgroup H is said to be S-supplemented in G if G has a subgroup T such that T ∩ H ⩽ HsG and HT = G; see [Skiba, J. Algebra 315: 192–209, 2007].