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Beginning with Hebcal for Unix 3.9, Asara B’Tevet is allowed to fall on a Friday. This is a significant bugfix.

According to http://www.ou.org/chagim/roshchodesh/tevet/fast.htm

If a public fast occurs on Erev Shabbat, we fast the entire day till the conclusion of the fast, even though it means entering Shabbat while fasting. Neither “Avinu Malkeinu” nor “Tachanun” are recited at Mincha.  The fast continues until after the completion of Maariv for Shabbat (after the appearance of the stars).  One should not eat or drink anything until after Kiddush.

Nowadays our calendar calculation is such that the only public fast which can fall on Erev Shabbat is the tenth of Tevet.

Older versions of hebcal incorrectly moved a Friday Asara b’Tevet fast to Thursday. We apologize for the inconvenience.