If we adopt a child from our friends, is her name was registered by the name of his real father or adopted father

Well I think that if you adopt the child then he/she may abandon her name because the child is now under your care.

Adoption is not Allowed in ISLAM, but, one can take care of anyone following Shariyat rules means Parda/hijaab etc. One can utilize love, money, and all capabilities to take care of anyone. But, Adopting is different. Because, Adopted child is not your "mehram" with which you can interact freely as with "Mehrams". Hence it is not allowed, but Taking care is totally allowed, condition is that you should use the way of Shariyat.