It is customary after a Torah learning session to recite "Rabbi Hananya Ben Akashya Omer…" followed by Kaddish Al Yisrael. This Kaddish recitation is especially significant, and in fact the Gemara comments that "the entire world stands" on the Kaddish recited after learning Aggadah. Our community also follows the custom of the Arizal (Rav Yishak Luria of Safed, 1534-1572) to recite half-Kaddish just before beginning Arbit. We recite the three Pesukim of "Hashem Seva-ot Imanu Misgab Lanu Elokeh Yaakob Selah," "Hashem Seva-ot Ashreh Adam Bote’ah Bach," "Hashem Hoshi’a Ha’melech Ya’anenu Be’yom Kor’enu," followed by half-Kaddish, after which we proceed to "Ve’hu Rahum" and "Barechu." Although many Ashkenazic communities do not follow this custom, our practice follows the teachings of the Arizal that the recitation of these verses and the half-Kaddish is part of the Tikkun (spiritual "rectification") that we seek to achieve through the Arbit service.
The question arises as to the proper procedure to follow if a Torah learning session concludes immediately before Arbit. In many synagogues, a Shiur is delivered right before Arbit, and the question is whether the Kaddish Al Yisrael recited after the class can serve also as the introductory Kaddish before Arbit, or if two separate Kaddishim should be recited.
The Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939), while not addressing this question, notes (260:227) that the custom in Yeshivat Bet-El in Jerusalem was to read Shir Ha’shirim, followed by Kaddish, immediately before Arbit on Friday night, and they would begin Arbit with a second Kaddish. This would appear to provide a precedent for requiring two separate Kaddishim, rather than have a single Kaddish recitation serve two different purposes. Just as the Kaf Ha’haim requires separate Kaddishim for Shir Ha’shirim and for Arbit, seemingly, a congregation that completes a Shiur before Arbit should likewise recite one Kaddish for the learning and a second Kaddish to introduce Arbit.
On the other hand, one could argue that on Friday night the Kaddishim serve special purposes related to the "Neshama Yetera" ("additional soul") that we receive through the Friday night service. If so, then we cannot necessarily draw any conclusions from the Kaf Ha’haim’s comments with regard to the situation of a Torah class which concludes right before Arbit.
Rav Rahamim Shayo (contemporary) discusses this issue in his Mehkereh Eretz, and he contends that each Kaddish serves a separate function, and thus two Kaddishim should be recited. He notes that this is, in fact, the custom in the Yeshiva where he studies, Yeshivat Nahar Shalom – to recite Kaddish Al Yisrael after the Shiur, followed by the three Pesukim of "Hashem Seva-ot" and half-Kaddish to begin Arbit. Rav Shayo relates that when this question was posed to Hacham Bension Abba Shaul (Israel, 1923-1998), he replied that a single Kaddish may be used for both purposes, but Rav Shayo understood this response to mean that a congregation may recite just one Kaddish, but can also recite two if it so chooses. Furthermore, Hacham Bension recommended bringing this question to Rav Mordechai Sharabi, Rosh Yeshiva of Nahar Shalom, where, as mentioned, two separate Kaddishim are recited.
It turns out, then, that both options are perfectly acceptable, and a congregation can choose either to recite a single Kaddish or two separate Kaddishim in this case. It must be emphasized, however, that if they choose to recite just one Kaddish, they must still recite the three Pesukim of "Hashem Seva-ot Imanu…" and should not proceed directly to "Ve’hu Rahum" after Kaddish Al Yisrael.
Summary: If a Torah class ends right before Arbit, and Kaddish Al Yisrael is recited (as is the custom), the congregation can choose either to recite an additional half-Kaddish before "Ve’hu Rahum" at the beginning of Arbit, or to skip that Kaddish, in light of the fact that Kaddish Al Yisrael had just been recited. Even in the latter case, however, they must ensure to recite "Hashem Seva-ot Imanu…"