Intermediately operator preserves quotient-transitivity

Statement
Suppose $$p$$ is a fact about::quotient-transitive subgroup property. Let $$I$$ denote the intermediately operator. Then, the property $$I(p)$$ is also a quotient-transitive subgroup property.

Applications

 * Intermediate characteristicity is quotient-transitive
 * Intermediate full invariance is quotient-transitive