Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure

Statement
Suppose $$p$$ is a subgroup property satisfying the fact about::intermediate subgroup condition: in other words, if $$H \le K \le G$$ and $$H$$ has property $$p$$ in $$G$$, then $$H$$ has property $$p$$ in $$K$$.

Then, the following are equivalent:


 * The property $$p *$$ Normal (where $$*$$ denotes the fact about::composition operator), i.e., the property of being a subgroup with property $$p$$ of a fact about::normal subgroup. $$H$$ has the property $$p *$$ Normal in $$G$$ if there exists a normal subgroup $$K$$ of $$G$$ such that $$H$$ has property $$p$$ in $$K$$.
 * The property of satisfying property $$p$$ inside the fact about::normal closure. In other words, $$H$$ has this property in $$G$$ if $$H$$ satisfies property $$p$$ as a subgroup of its normal closure in $$G$$.

Examples
This fact gives a number of equivalent formulations of subgroup properties obtained by composing some subgroup property with normality, specifically: