First of all, if a child is adopted and we are unsure of the child's birth
religion, the child must be converted in a halachic conversion. The child
must then confirm the conversion at halachic adulthood (bar/bat mitzvah) so
that there is no question that they intend to remain Jewish. If the
adoptee is a female and she was converted halachically, she is Jewish,
period. If not, then her children are not Jewish no matter which place of
worship they were raised with. It doesn't matter if the husband is Jewish
or not, as Judaism is passed on by the mother.