Analysis of ion behavior in acidic media is as follows: 1. \( \text{MnO}_4^- \) stability: \( \text{MnO}_4^- \) remains stable and retains its purple hue in acidic conditions due to a lack of further reactions. 2. \( \text{MnO}_2^{2-} \) behavior: \( \text{MnO}_2^{2-} \) disproportionates in acid, yielding purple \( \text{MnO}_4^- \) and colorless \( \text{Mn}^{2+} \). 3. \( \text{CrO}_4^{2-} \) conversion: \( \text{CrO}_4^{2-} \) transforms into \( \text{Cr}_2\text{O}_7^{2-} \) under acidic conditions, causing a color change from yellow to orange due to equilibrium shifts. 4. \( \text{FeO}_4^{2-} \) decomposition: \( \text{FeO}_4^{2-} \) breaks down into \( \text{Fe}^{3+} \), which is either colorless or pale yellow based on concentration. Consequently, \( \text{MnO}_4^- \) is the species among those listed that maintains its purple color in an acidic medium.
Final Answer: \[ \boxed{\text{MnO}_4^-} \]