Praise be to Allah the Lord of the Worlds.
Taking care of an orphan is one of the greatest good deeds that bring a Muslim closer to Allah the Almighty because the Messenger of Allah (PBUH) said: "He who takes care of an orphan, whether he is his relative or a stranger, will be in Jannah with me like these two". The narrator, Malik bin Anas raised his forefinger and middle finger for illustration." {Muslim}.
Taking care of the aforementioned girl is urged by the tolerant teachings of Islam reflecting the greatness of this religion in terms of achieving social solidarity and promoting the spirit of love in society taking into considerationthe fact that Islam doesn`t stipulate that the sponsored orphan should be a Muslim. Rather, the principle of Ihsan (Doing one`s best to say and do only what Pleases Allah Almighty and conforms to His Commands.) rests on viewing people as human beings regardless of belief or religious background. In addition, since one receives a reward for practicing Ihsan towards plants and animals, it is all the more valid to practice it with orphans or illegitimate children.
According to scholars, taking care of an illegitimate child entails the same reward as taking care of an orphan. Al-Shirbin said: "Scholars are of the view that the word "Illegitimate child" falls under the meaning of the word "Orphan" although the former isn`t called "An orphan".
In conclusion, it is permissible to take care of an illegitimate girl in a non-Muslim country under the condition that the care taker doesn`t give her his name, because she is a stranger to him. Therefore, she takes the rulings of his non-Mahrams. However, there is nothing wrong with her calling him father in the sense of sponsorship not lineage. He should also tell her that he isn`t her real father. And Allah the Almighty Knows Best.