Intermediately isomorph-conjugate of normal implies pronormal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., intermediately isomorph-conjugate subgroup of normal subgroup) must also satisfy the second subgroup property (i.e., pronormal subgroup)
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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties , to another known subgroup property
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Statement

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.

Related facts

Weaker facts

• Intermediately isomorph-conjugate 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