What is the ruling on my false testimony that so-and-so is my relative's maternal aunt to facilitate her arrival for Hajj, and is an expiation for a broken oath (kaffarat yameen) incumbent upon me?

Bearing witness that the aforementioned woman is the paternal aunt (khāla) of the man, knowing that this is contrary to the truth, is a false statement and false testimony. Allah the Almighty condemned this, saying: "So avoid the uncleanliness of idols and avoid false statement" (22:30). And in describing the believers, He said: "And those who do not bear witness to falsehood" (25:72). The Prophet, peace and blessings be upon him, described it as one of the gravest major sins. As for saying that the woman is the sister of his mother from her father and mother, with the intention of referring to the progeny of Adam and Eve, this is a form of tawriya (equivocation). Tawriya is a permissible alternative to lying, but tawriya is not valid in testimony related to rights. If you did not swear an oath on what you testified to, there is no expiation other than repentance and regret. If you did swear an oath, it is a yamīn ghamūs (a perjured oath deliberately sworn to deceive), for which there is no expiation according to the majority of scholars except repentance. It is more cautious to offer an expiation for an oath according to some scholars, while noting that expiation does not suffice for repentance.