Answer: 4) Either my sister or my mom are going to pick up groceries.
Explanation:
Sister and Mom are pronouns, which means that the verb is going to be relevant to the pronouns. They aren't a topic. When she says they are going to pick up groceries, there would be an agreement because groceries are something you pick up; therefore, that verb is going to agree with the subject.
So, "either my sister or my mom are going to pick up groceries" is going to be your correct answer.
