For \(\mathrm{AuF}_{3}\): Cation: \(\mathrm{Au}^{3+}\), Anion: \(\mathrm{F}^{-}\) \\
For \(\mathrm{AuBr}\): Cation: \(\mathrm{Au}^{+}\), Anion: \(\mathrm{Br}^{-}\) \\
For \(\mathrm{MnO}_{2}\): Cation: \(\mathrm{Mn}^{4+}\), Anion: \(\mathrm{O}^{2-}\) \\
For \(\mathrm{CrI}_{3}\): Cation: \(\mathrm{Cr}^{3+}\), Anion: \(\mathrm{I}^{-}\)
Thus, the final boxed answers are:
\[
\boxed{\text{For } \mathrm{AuF}_{3}: \mathrm{Au}^{3+}, \mathrm{F}^{-}}
\]
\[
\boxed{\text{For } \mathrm{AuBr}: \mathrm{Au}^{+}, \mathrm{Br}^{-}}
\]
\[
\boxed{\text{For } \mathrm{MnO}_{2}: \mathrm{Mn}^{4+}, \mathrm{O}^{2-}}
\]
\[
\boxed{\text{For } \mathrm{CrI}_{3}: \mathrm{Cr}^{3+}, \mathrm{I}^{-}}
\]