Who is responsible for returning the value of the coupons to the institution: the one who benefited from them, or the one who obtained them and gave them to others, knowing that she acquired them based on incorrect information?

First, providing false information about one's salary is a forbidden lie and an unlawful taking of money, unless there is a necessity that permits lying to obtain essentials.

Second, if you are not compelled, it is not permissible for you to take the money if the institution stipulates that the salary must not exceed a certain limit. As for guaranteeing the money, if the institution grants aid to those whose father's actual salary is what yours is and you are eligible, then you are not required to guarantee it, but repentance from lying is obligatory. If the institution does not grant aid to those whose father's salary is what yours is, then you must return the money to it by any means.