Communities of Georgian origin, whether abroad or in Israel, customarily read and study the Avot tractate before the Mincha prayer during the six Sabbath days between the festivals of Passover and Atzeret, in keeping with the Sephardim's custom.
Some of our Ashkenazi brothers have the custom of continuing until the month of Tishrei (in addition to between Passover and Atzeret), until the Sabbath before Rosh Hashanah, so that as a result they study Pirkei Avot four times. We might astutely remark that it is certainly preferable to read Pirkei Avot, even after the Festival of Shavuot, rather than to hold idle conversation, but our custom, as noted above and in keeping with the custom of the sons of Sepharad, is to read Pirkei Avot from Passover to Atzeret. One reads during the Sabbath close to Shavuot - the time our Torah was given – and those who customarily read Pirkei Avot during the entire summer Sabbath days acknowledge that the six Sabbath days between Passover and Atzeret are the principal period of the custom.