In Talmudic times, it was customary for the Hazan or Rabbi to recite the complete Kiddush in the synagogue after Arbit on Friday night. The reason was that guests from other communities would spend Shabbat in the synagogue. There were rooms to the side of the sanctuary for guests, and food was made available for them. Therefore, in order to ensure they would hear Kiddush, the Hazan or Rabbi would recite Kiddush on their behalf after the Arbit service. Since the guests would eat in the synagogue, the recitation of Kiddush was done in fulfillment with the requirement of "Kiddush Bi’mkom Se’uda" (that Kiddush be recited in the place where one eats).
Today, of course, no guests spend Shabbat in the synagogue, as everyone in the synagogue hers Kiddush and eats the Shabbat meal either at home or as guests in other people’s homes. Therefore, as Hacham Ovadia Yosef writes in Hazon Ovadia – Shabbat (beginning of vol. 2), Kiddush should not be recited in the synagogue on Friday night nowadays. The rare exception to this rule is a situation where there are people in the synagogue who are unknowledgeable about Halacha and will not be reciting or hearing Kiddush that night. If a congregation knows that such people are in attendance, then the Rabbi should recite Kiddush on their behalf. Even though they will not be eating in the synagogue, and will thus not fulfill the obligation of "Kiddush Bi’mkom Se’uda," nevertheless, Kiddush should be made for them so they can at least fulfill the basic Torah obligation to hear Kiddush. (The Torah obligation to recite Kiddush is normally fulfilled through the recitation of Arbit, but those who are unknowledgeable and will not be reciting Kiddush presumably did not recite Arbit, either, and thus did not fulfill their Torah obligation.) The Rabbi who recites Kiddush can either give a minor some wine to drink, or can drink a Rebi’it himself and thereby fulfill the requirement of "Kiddush Bi’mkom Se’uda."
Another exception to this rule is a case where a communal meal or social function is being held in the synagogue, in which case, of course, Kiddush should certainly be recited, as the meal will be eaten there. As a general rule, however, Kiddush should not be recited in the synagogue after Arbit on Friday night.
Summary: Although in Talmudic times Kiddush was recited in the synagogue on Friday nights after the Arbit service, this is not done today, except in the rare case when there are people in the synagogue who are unknowledgeable and will not be reciting or hearing Kiddush after Arbit.
