In the given situation, since the sacrifice was made on his behalf with his permission by buying a separate animal for him, this sacrifice is valid, and in this case the owner of the sacrificial meat is also the person on whose behalf the sacrifice was made. Also, since a separate animal was bought for him with his permission, the price of that animal will also be obligatory on him, however, if the person who made the sacrifice waives this price, then it is also valid.
The ruling of all the situations mentioned in the clarification is as follows:
1. If the obligatory sacrifice is made on behalf of the other person without his permission, then this sacrifice is not valid at all.
2. If the obligatory sacrifice is made on behalf of the other person with his permission, then there are two cases:
a) If no separate animal was bought for the sacrifice, but rather he sacrificed his own pet animal or included his pet animal, then this sacrifice is also not valid, because this is a case of "gift" and for a gift to be valid, possession is necessary, and here possession was not found.
b) If a separate animal was bought for the sacrifice, or an animal was bought and included in it, then this sacrifice will be valid. And in this case the owner of the meat will be the person on whose behalf the sacrifice is being made, so it is necessary to deliver the meat to him, however if he does not want to take the meat then it will be a donation from him. Also, since the animal was bought with his permission, the price of that animal will also be obligatory on him, however, if the person who made the sacrifice does not want to take this price, then there is also room for that.
3. If a nafil sacrifice is made on behalf of the other person, then such a sacrifice will be valid in any case, whether it is with his permission or without his permission, and in such a case the owner of the sacrificial meat will be the person who made the sacrifice. The person on whose behalf the sacrifice is made is not required to deliver the meat.