One of the points during Kaddish when the congregation answers "Amen" is after "Shemeh De'Kudsha Berich Hu." Ashkenazic custom, however, is not to recite "Amen" at that point, and to recite "Berich Hu" instead of "Amen."
Rav Moshe Feinstein (1895-1986) rules that according to the Ashkenazic custom, a person who hears Kaddish while reciting Pesukeh De'zimra should not respond "Berich Hu." Although he may answer "Amen" to Kaddish, he may not answer "Berich Hu," as this does not qualify as an obligatory response that warrants interrupting Pesukeh De'zimra. This position was taken also by Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995).
Rav Yaakov Kamenetsky (1891-1986) and Rav Chaim Kanievsky (1928-2022) went even further, ruling that Ashkenazim should preferably not answer "Berich Hu" to the Hasi-Kaddish recited before the Amida prayer during Arbit. Interruptions are not permitted during Arbit from Barechu through the Amida prayer, though "Amen" responses are allowed. According to the aforementioned Poskim, the response of "Berich Hu" is not significant enough to warrant an interruption before the Amida, and so the congregation should not respond "Berich Hu." Rav Moshe Feinstein, however, maintained that the "Berich Hu" response is allowed at that point.