Is the necklace my friend bought permissible or forbidden, given that her husband's father helped him purchase it with a portion of gold he owns, around which there are doubts that it is not permissible? Is it permissible for her to donate a portion of it so that the remainder becomes permissible, or should she give it all away in charity, or should she return it to the husband's father?

As long as the matter is merely rumors, and the man denies the accusation, the default is to believe him and consider the gold his property. This is because the fundamental principle regarding a Muslim is integrity, and their dealings are presumed to be valid. Therefore, there is no harm for the woman in benefiting from this gold, and she is not obligated to donate any part of it, nor to return it to her husband's father.