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Using a sample of 4.48×108 𝜓(3686) events collected with the BESIII detector at the BEPCII collider, we study the two-photon decays of the pseudoscalar mesons 𝜋0, 𝜂, 𝜂′, 𝜂(1405), 𝜂(1475), 𝜂(1760), and 𝑋(1835) in 𝐽/𝜓 radiative decays using 𝜓(3686)→𝜋+𝜋−𝐽/𝜓 events. The 𝜋0, 𝜂, and 𝜂′ mesons are clearly observed in the two-photon mass spectra, and the branching fractions are determined to be 𝐵(𝐽/𝜓→𝛾𝜋0→3𝛾)=(3.57±0.12±0.16)×10−5, 𝐵(𝐽/𝜓→𝛾𝜂→3𝛾)=(4.42±0.04±0.18)×10−4, and 𝐵(𝐽/𝜓→𝛾𝜂′→3𝛾)=(1.26±0.02±0.05)×10−4, where the first error is statistical and the second is systematic. No clear signal for 𝜂(1405), 𝜂(1475), 𝜂(1760) or 𝑋(1835) is observed in the two-photon mass spectra, and upper limits at the 90% confidence level on the product branching fractions are obtained.