All the laws of Shabbat apply equally to both men and women, and therefore, women are included in the Torah obligation of Kiddush just like men.
The Halachic authorities addressed the question of how a husband can recite Kiddush on behalf of his wife on Friday night after he prayed Arbit. The Torah obligation of Kiddush is introduced with the command, "Zachor Et Yom Ha’Shabbat Le’kadesho," which refers to verbally mentioning Shabbat. On the level of Torah obligation, one needs simply to verbally speak of the sanctity of Shabbat; it was Hazal who then established the requirement to recite the Kiddush text over a cup of wine. Therefore, once a man recites Arbit on Friday night, which includes mention of the Kedusha of Shabbat, he has fulfilled his Torah obligation, and now has only the Rabbinic requirement to recite Kiddush over wine. Most women, however, do not recite Arbit, and thus when a husband returns home from the synagogue on Friday night, he bears a Rabbinic obligation of Kiddush while the wife bears a Torah obligation. The question thus arises as to whether the man, who bears a lower level of obligation than his wife does, may recite Kiddush on her behalf so she can fulfill her Misva.
The consensus among the Halachic authorities is that a husband can recite Kiddush on his wife’s behalf, because of the principle of "Kol Yisrael Arebim Zeh La’zeh" – "All Jews are responsible for one another." This means that every Jew bears responsibility not only for his own observance of Misvot, for also for that of his fellow Jews. On the basis of this rule, the Gemara in Masechet Rosh Hashanah establishes that "Af Al Pi She’yasa Mosi" – even after a person has fulfilled his obligation, he can still perform the Misva on behalf of somebody else. The classic example is that of a person who already heard Shofar blowing on Rosh Hashanah, and thus fulfilled his obligation, but may nevertheless blow the Shofar for somebody else who has yet to fulfill the Misva. Since we are all responsible for each other’s Misva observance, even after one has fulfilled the Misva of Shofar he is still eligible to blow the Shofar for others. By the same token, even though a husband fulfills his Torah obligation of Kiddush by reciting Arbit, he may nevertheless recite Kiddush for his wife so she can fulfill her Torah obligation. Although the Dagul Me’rebaba (Rav Yechezkel Landau of Prague, 1713-1793) questioned whether the principle of "Kol Yisrael Arebim Zeh La’zeh" applies to a man fulfilling Misvot for a woman, Rabbi Akiva Eger (1761-1837) proves (in his glosses to the Shulhan Aruch, 271) that the rule indeed applies to such situations.
Therefore, as established by numerous Halachic authorities – including the Aruch Ha’shulhan (271:5-6) and Hacham Ovadia Yosef (Hazon Ovadia – Shabbat, vol. 2) – a husband is certainly allowed to recite Kiddush for his wife, despite the fact that he has recited Arbit and she has not. Hacham Ovadia adds that this ruling applies in the opposite case, as well. If a woman recited Arbit and the husband did not, the wife may recite Kiddush for her husband.
Summary: Despite the fact that men fulfill the Torah obligation of Kiddush on Friday night by reciting Arbit, and most women do not recite Arbit, nevertheless, a woman can fulfill her Kiddush obligation on Friday night by listening to her husband’s recitation.
