Well, the dual citizenship sometimes is complicated.
First of all, if an ancestor who left italy to go abroad was born before 1860, he can not inherit the italian citizenship for his descendants.
The law about italian citizenship and right to acquire it is dated 1912, so everyone who naturalized in the new country were emigrated, before 1912, can not inherit it to his grandchildren.
During those times the birth, marriage and death form record didn't contain the information about the citizenship of the people of the record.
In fact sometimes italian consulate abroad would like to have a statement from the civil record office were the italian ancestor was born, that he was born as italian citizenship.
Starting from the point that this kind of statement is not required by the italian citizenship directive, usually the registrar don't issue this kind of statement because there is not written anywhere that the ancestor was born as italian citizen.
This information is stated on the records kept at Population registry office (Ufficio Anagrafe), this office is founded around 1930 in every single municipality of italy.
But the consulate try very hard to have it because for sure a baby born in italy from polish or irish or german parents, is not italian, italian citizenship is by blood, not because the baby was born on the italian territory.
If a baby is born from an italian parent and from a non italian parent, the baby is italian.
I hope you got the idea now.
To answer your question, if an italian father didn't recognize the baby, and the non italian mother recognized the baby, it is not italian.
I do not understand how a mother put the last name of the father of the baby if his father disappeared.
Today if a baby is born, and both parents are not recognizing him, so it is in the state of abandoned baby, and put for adoption, it doesn't matter if the baby is black, yellow, or anything, the law states he is an italian citizen.
I hope I was clear, and if you think you have more questions you know where to find me