Collection of groups satisfying a strong normal replacement condition

Definition
Suppose $$\mathcal{S}$$ is a finite collection of finite $$p$$-groups, i.e., groups of prime power order where the prime is $$p$$. We say that $$\mathcal{S}$$ satisfies a strong normal replacement condition if it satisfies the following equivalent conditions:


 * 1) For any finite $$p$$-group $$P$$ that contains a subgroup $$H$$ isomorphic to an element of $$\mathcal{S}$$, $$P$$ contains a normal subgroup $$K$$, also isomorphic to an element of $$\mathcal{S}$$, such that $$K$$ is contained in the normal closure of $$H$$ in $$P$$.
 * 2) For any finite $$p$$-group $$P$$ that contains a 2-subnormal subgroup $$H$$ isomorphic to an element of $$\mathcal{S}$$, $$P$$ contains a normal subgroup $$K$$, also isomorphic to an element of $$\mathcal{S}$$, such that $$K$$ is contained in the normal closure of $$H$$ in $$P$$.# For any finite $$p$$-group $$Q$$ and normal subgroup $$P$$ of $$Q$$, if there exists a subgroup of $$P$$ isomorphic to an element of $$\mathcal{S}$$, there exists a subgroup of $$P$$ that is normal in $$Q$$ and is isomorphic to an element of $$\mathcal{S}$$.

Stronger properties

 * Weaker than::Collection of groups satisfying a universal congruence condition

Weaker properties

 * Stronger than::Collection of groups satisfying a weak normal replacement condition