Intermediately isomorph-conjugate of normal implies pronormal

Property-theoretic statement
The subgroup property of being an intermediately isomorph-conjugate subgroup of normal subgroup (i.e., the subgroup property obtained by applying the composition operator to the properties intermediately isomorph-conjugate subgroup and normal subgroup) is stronger than the subgroup property of being a pronormal subgroup.

Verbal statement
Any intermediately isomorph-conjugate subgroup of a normal subgroup of a group is pronormal.

Weaker facts

 * Intermediately isomorph-conjugate implies pronormal
 * Procharacteristic of normal implies pronormal

Corollaries

 * Sylow of normal implies pronormal
 * Frattini's argument is a slight weakening of this.

Similar facts

 * Intermediately automorph-conjugate of normal implies weakly pronormal
 * Intermediately normal-to-characteristic of normal implies intermediately subnormal-to-normal

Converse
Most natural choices of converse to this statement aren't true. Specifically, it is not necessary that if $$H \le G$$ is such that whenever $$G$$ is normal in $$K$$, $$H$$ is pronormal in $$K$$, then $$H$$ is intermediately isomorph-conjugate in $$G$$. What we can guarantee for $$H$$ is that it is a procharacteristic subgroup of $$G$$.

Facts used

 * 1) uses::Intermediately isomorph-conjugate implies procharacteristic
 * 2) uses::Procharacteristic of normal implies pronormal

Proof
The proof follows by combining facts (1) and (2).