p-Hypercyclically embedding and ICΦ-subgroups of finite groups

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text]. In this paper, we shall obtain some criteria for hypercyclically embedding and [Formula: see text]-hypercyclically embedding of normal subgroups of [Formula: see text] under the assumption that some subgroups of [Formula: see text] of prime power order are [Formula: see text]-subgroups of [Formula: see text].