How do parents add the name of a child who is not theirs to official records when raising him, knowing that adoption is forbidden?

If an adopted child has a known lineage, they should be attributed to it. If they do not have a known lineage, they can be generally attributed by saying: "So-and-so, son of Abdullah," or "son of Abdurrahman," to remove hardship and achieve the purpose, while avoiding the mixing of lineages and the loss of rights. Legal authorities can be consulted for assistance in facilitating this.