Can the sheep bought by the father-in-law be considered an 'aqiqah for the newborn?

If your father-in-law donates his sheep to the child's father, or to whoever is responsible for his maintenance, or if he permits him to slaughter it as an 'Aqiqah for his son, then it is considered a valid 'Aqiqah. The donation of an 'Aqiqah by someone other than the father is not prohibited. The one who is obligated to perform it is he who is responsible for the newborn's maintenance, and no one else should perform it without his permission. However, if your father-in-law slaughters the sheep on his own initiative and without the permission of the child's guardian, then that is not considered an 'Aqiqah and does not suffice for it.