Branching fraction measurements of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>η</mml:mi><mml:mi>η</mml:mi></mml:math>

Using a sample of $1.06\ifmmode\times\else\texttimes\fi{}{10}^{8}$ ${\ensuremath{\psi}}^{\ensuremath{'}}$ decays collected by the BESIII detector, ${\ensuremath{\chi}}_{c0}$ and ${\ensuremath{\chi}}_{c2}$ decays into ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ and $\ensuremath{\eta}\ensuremath{\eta}$ are studied. The branching fraction results are $\mathrm{Br}({\ensuremath{\chi}}_{c0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0})=(3.23\ifmmode\pm\else\textpm\fi{}0.03\ifmmode\pm\else\textpm\fi{}0.23\ifmmode\pm\else\textpm\fi{}0.14)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, $\mathrm{Br}({\ensuremath{\chi}}_{c2}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0})=(8.8\ifmmode\pm\else\textpm\fi{}0.2\ifmmode\pm\else\textpm\fi{}0.6\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, $\mathrm{Br}({\ensuremath{\chi}}_{c0}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\eta})=(3.44\ifmmode\pm\else\textpm\fi{}0.10\ifmmode\pm\else\textpm\fi{}0.24\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and $\mathrm{Br}({\ensuremath{\chi}}_{c2}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\eta})=(6.5\ifmmode\pm\else\textpm\fi{}0.4\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, where the uncertainties are statistical, systematic due to this measurement, and systematic due to the branching fractions of ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\chi}}_{cJ}$. The results provide information on the decay mechanism of ${\ensuremath{\chi}}_{c}$ states into pseudoscalars.