All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Allaah forbade adoption which was common in the pre-Islamic era of ignorance and ordered to trace the child back to his (real) father. Allaah Says (what means): {And He has not made your claimed [i.e. adopted] sons your [true] sons. That is [merely] your saying by your mouths, but Allaah Says the truth, and He guides to the [right] way. Call them by [the names of] their fathers; it is more just in the sight of Allaah. But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you. }[Quran 33:4-5] For more benefit, please refer to Fatwa 123475.
However, if what is meant by adoption is just looking after and nurturing the child without tracing him back to anyone else than his (real) father, then this is permissible and this is one act among the acts of righteousness that Allaah likes. For more benefit, please refer to Fataawa 86312 and 84422.
As regards the ruling on breastfeeding, then if you breastfeed this child five times or more before he/she is two years old, then this child becomes your foster child and your husband becomes his/her suckling foster-father, and your children will become his/her suckling foster-brothers [or sisters]. For more benefit, please refer to Fatwa 91172.
Allaah Knows best.