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We present measurements of the $B\ensuremath{\rightarrow}\ensuremath{\eta}K\ensuremath{\gamma}$ branching fractions and upper limits for the $B\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}K\ensuremath{\gamma}$ branching fractions. For ${B}^{+}\ensuremath{\rightarrow}\ensuremath{\eta}{K}^{+}\ensuremath{\gamma}$ we also measure the time-integrated charge asymmetry. The data sample, collected with the BABAR detector at the Stanford Linear Accelerator Center, represents $232\ifmmode\times\else\texttimes\fi{}{10}^{6}$ produced $B\overline{B}$ pairs. The results for branching fractions and …