Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Udhiyah (an animal that is sacrificed for The Sake of Allah in the days of Eid Al-Adha) is a confirmed Sunna (Sunna Mu’akkadah) for the one who is financially able—that is, someone who possesses an amount exceeding their needs for the day and night of Eid and the days of Tashreeq, sufficient to purchase a sacrifice. It is considered a communal Sunna (Sunna Kifayah) for the members of one household, if their expenses are covered by a single person. If one of them offers the Udhiyah—even if they are not financially responsible for the others, such as the wife or one of the children—it fulfills the requirement for the household. However, the reward applies only to the one who performs the sacrifice, similar to how the obligation of funeral prayer is lifted when some people perform it, but others do not receive the reward unless the sacrificer intends to share the reward with them.
In this regard, Al-Imam Suleiman Al-Jamal (may Allah have mercy on him) said: "His statement: 'Sacrificing (Udhiyah) is Sunna'—that is, for a capable, free Muslim, whether completely or partially free. By 'capable' is meant someone who owns more than what he needs for the day and night of Eid and the days of Tashreeq, in an amount sufficient to purchase a sacrificial animal—contrary to those who disagreed and said it should only be in excess of his needs for the day and night [of Eid]. He must also be of sound judgment. Moreover, his statement: "If the household members are multiple'—meaning if their financial maintenance is the responsibility of one person, even if the homes are multiple. Then if one of them performs the sacrifice, even if it is someone who is not financially responsible (e.g. the wife or a child), the obligation is lifted from the rest, although the reward is specific to the one who performs the sacrifice. So the meaning of it being a communal Sunna, despite it being recommended for each of them individually, is that the obligation of recommendation falls away when someone else performs it, but the reward is not shared unless the sacrificer intends to include them in the reward—just like the funeral prayer. Yes, the author mentioned in his commentary on Muslim that if one includes others in the reward of the sacrifice, it is permissible." [Fateh Al-Wahhab/Vol. 5/P.251].
In conclusion, the Udhiyah is a confirmed Sunna for the one who is financially able. It is considered a communal Sunna for a single household—if one of them performs it, even if he or she is not the primary financial provider for the household, the recommendation is lifted from the rest. The one who offers the sacrifice may include whomever they wish in its reward. And Allah The Almighty Knows Best.