To Help A Grandmother Correct Her Grandson

Hello Everyone!
I need to sort out a situation affecting my extended family. Please help me.
Background: This is about my FIL's older brother's family. He, his wife, their 2 sons, DILs and their kids stay on a floor of a building. They own the entire floor. Although the 3 apartments on the floor are intact and are "assigned" to the father and the 2 sons each, the entire family spends most of the day together in the older couple's apartment. They cook and eat together. The brothers as well as their wives are very amicable and share a good relation with each other. (They may have had their own share of tiffs but nothing worth the mention.)
Situation: The 2 DILs are housewives and contribute to the household chores equally. The older one has a son (a new teenager) and the younger one has a daughter. These kids were born only a couple of years apart and are very fond of each other. The boy is average in studies, extra curricular activities, etc. but the girl is an overachiever. The boy is closer to the grandfather who pampers him a lot with attention, stories, gifts, etc. The girl is close only to her mother who is very intelligent and compassionate and very rational in her behavior.
Problem (as identified and iterated discreetly to me by the grandmother) : The boy and his mother have started envying the girl's success for a few years now. They do not even congratulate the girl or appreciate any of her achievements. Although the girl's mother has noticed this change, she refuses to acknowledge it in any way. The boy and his mother have started lying about the boy's achievements- marks scored in exams, success in any competitions, etc. (A) The boy is a serial liar who takes anyone and everyone for a ride with his tales. His mother is a party to it and has been caught red handed on multiple occasions till now (but not confronted anytime). (B) In these online times, he is constantly watching videos on the net or playing games and when someone enters their room, he quickly closes that window and switches to a study related window. He does the same if he is asked to sit in the family room. (C) The height was when the grandmother caught a pilferage. The grandmother had asked the grandfather for some cash and he had placed it on the desk in their room. The boy flicked 1 big note out of the small bundle. He had initially refused to admit this but on continuous questioning by the grandmother, he finally confessed it all. He also added that he has been picking cash from his father's wallet for some time now and that he shares the cash with his own mother!

The grandmother wants earnestly to correct this boy and bring him back on the right track without affecting the peace and harmony at home and without hurting anyone. She is also worried that the girl may walk on his footsteps in the future. She has asked me for help in anyway to set things right without spoiling my equation with any of her family members. I want to help her/ them but I don't know what to do.