Is this a merit scholarship or need based financial aid? If it’s merit scholarships, some schools have a no appeal policy so you would need to check on that.
Are college A and B peer institutions? If not, you can probably stop right here. It’s typical for less competitive schools to offer more aid to attract top students.
You don’t need permission from college A to talk about their offer.
If it’s merit based aid - that is usually through the admissions office. If it’s is need based aid, then the financial aid office.
For both kinds of appeals, you need a reason why you are coming back to the school. If it’s to reconsider merit aid, have the student’s grades improved? Won a new prestige award? Recognized for something that wasn’t on the common app? If for financial aid, has there been a change in income?