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The rule is that an apostrophe followed by the letter "s" is only used to indicate possession with nouns, not with pronouns (rather than prepositions). "Its" is a pronoun just like "his", "hers", "theirs", "yours", "ours", and "mine". There should be no apostrophes in any of them.

However, I think the author may have been referring to two different mistakes here. People sometimes do put apostrophes in plurals, which is also incorrect. It is a bit confusing to have the subtitle and the text refer to two different mistakes.