All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If this woman is in need and has no one among her relatives to spend on her, then the Muslim community must take care of her, and not leave her to the point that she is lost or perished. Allah Says (what means): {And does not encourage the feeding of the poor.} [Quran 107:03]
Al-Qurtubi said in his Tafseer (interpretation of the Quran): “The meaning of the statement is: “They do not do it if they are able to, and they do not encourage it if it is difficult for them to do it....” [End of quote]
Your help to this woman is religiously required, and you will be rewarded for it greatly. However, you are obligated to deal with her in accordance with the religious guidelines and to avoid everything that may lead to temptation because she is a non-Mahram to you. If it was possible to search for a righteous man to marry her, that would be a good thing.
On the other hand, we warn of the prohibition of adoption, as Allah Says (what means): {…And he has not made your adopted sons your [true] sons. That is [merely] your saying by your mouths, but Allah says the truth, and He guides to the [right] way.} [Quran 33:04]
The adopter must attribute the lineage of the child a true lineage if he knows the origin of his lineage, or attribute him/her a general lineage if he is unawre of his/her lineage. Allah Says (what means): {Call them by [the names of] their fathers; it is more just in the sight of Allah. But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you.} [Quran 33:04]
Allah knows best.