The above discussion is true only for a group of women exclusively. However, if women ate together with three men, the women are obligated to join the Zimun of the men. Mishna Berura (Beiur Halacha 199 s.v. Aval) points out that this obligation applies to even one woman who ate with three men. Since a woman is required to participate in the Zimun of three men, it follows that in such instances, the woman may not bentch on her own. Furthermore, if the woman is not at the table when the men wish to bentch, the men must ask her to join them in Zimun and wait until she returns to the table.
Igros Moshe (OC 5:9:10) writes that the requirement for women to be part of the Zimun is not the same on weekdays, as for Shabbos. During the week, the wife is often busy cooking, serving and taking care of her children, and the meal is typically eaten quickly. Therefore, the wife is not considered to be eating with the men, and she may bentch on her own and not participate in Zimun. (If the meal is not rushed and the wife eats with her family, she is required to join the Zimun.)
In contrast, on Shabbos, the family eats together, as people are less rushed and meals last longer. Therefore, on Shabbos, the wife is considered part of the group and she must participate in the Zimun, and the men may not bentch without her.
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