〜ふりをする
pretend to
Plain form + ふりをする (な-adj + な · Noun + の)
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ふりをする means pretending, putting on a show for others: 彼は寝たふりをしている, "he's pretending to be asleep". 知らないふりをした, "I pretended not to know".
It follows a plain form: a verb, an い-adjective, a な-adjective with な (平気なふり, "pretending to be fine"), or a noun with の (学生のふり).
With a た-form, the state is already in place: 寝たふり is "pretending to be asleep", not "pretending to have slept".
Compare つもり, which is what someone believes in their own head. ふり is always about deceiving others, even if only a little.
Examples
彼は寝たふりをしている。
He's pretending to be asleep.
知らないふりをした。
I pretended not to know.
忙しいふりをして、電話に出なかった。
I pretended to be busy and didn't answer the phone.
彼女は平気なふりをしていた。
She was pretending she was fine.
聞こえないふりをするのはやめて。
Stop pretending you can't hear me.