In addition to the recitations of Kaddish during the prayer service, there are also other occasions that warrant the recitation of Kaddish, namely, when ten or more men study Torah together. Upon the conclusion of the Torah study session, Kaddish should be recited. The exact recitation depends on the nature of the study. If a group assembled for the recitation of Tehillim, for example, then after the recitation they should recite Kaddish Yeheh Shelama. However, after the study of Torah She’be’al Peh, such as Mishna or Gemara, then the group recites Kaddish Al Yisrael Ve’al Rabbanan (known as "Kaddish De’rabbanan").
There is a widespread custom to recite at the conclusion of a Torah class the final Mishna of Masechet Makot: "Rabbi Hananya Ben Akashya Omer Rasa Ha’kadosh Baruch HU…" This custom is practiced even when a group studies Torah She’be’al Peh, in which case they must in any event recite Kaddish Al Yisrael. The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) explained this custom as based on a comment by the Gemara in Masechet Sota. The Gemara lists the different "pillars" upon which the world stands, one of which is the Kaddish recited after studying Aggada – the homilies and philosophical insights of the Sages. Although Kaddish should be recited after every Torah study session, there is special significance to the Kaddish recited after the study of Aggada. Therefore, it became customary to recite after every Torah class the Mishna of "Rabbi Hananya Ben Akashya Omer," a brief passage of Aggada, in order to be able to recite Kaddish after learning Aggada.
Several Poskim, including the Hid"a, Kaf Ha’haim, Ben Ish Hai and Mishna Berura, mention the Halacha of "En Marbim Be’kaddishim." This means that we should not try to create situations requiring extra recitations of Kaddish. Although we should certainly recite Kaddish when it is warranted, it is improper to go out of one’s way to facilitate an extra Kaddish. One situation where this Halacha applies is when Arbit is recited immediately following a Shiur. Upon the conclusion of the Shiur, "Rabbi Hananya Ben Akashya" should be recited, followed by Kaddish Al Yisrael. At that point, the Minyan should proceed directly to Arbit, with the recitation of "Ve’hu Rahum" and "Barechu." Sometimes, people in this situation first recite the verses of "Hashem Seva-ot Imanu" before Arbit to recite an additional Kaddish. This is incorrect; they should proceed directly to "Ve’hu Rahum" afterthe Kaddish is recited following the Shiur.
Summary: After a group studies Torah, Kaddish should be recited if a Minyan is present. It is customary to recite the Mishna of "Rabbi Hananya Ben Akashya Omer" after the study session, followed by Kaddish Al Yisrael. It is improper to specifically create situations to enable extra Kaddish recitations.
