How does ${H_2}{O_2}$ react with $KMn{O_4}$ in an alkaline medium?

Hint:
When hydrogen peroxide reacts with potassium permanganate in alkaline medium then, the potassium moves from potassium permanganate and combines with hydroxide and manganese oxide is left.

Complete step by step answer:
${H_2}{O_2}$ is the chemical formula of hydrogen peroxide. It is colourless and odourless liquid. The taste of hydrogen peroxide is bitter. It is completely miscible with water, alcohol and ether all in the same proportion. The colour of pure hydrogen peroxide is pale blue and is the thick syrupy liquid.
In acidic medium, the hydrogen peroxide can oxidize acidified ferrous sulphate to ferric sulphate and in neutral solution, it can oxidize lead sulphide to lead sulphate.
The dilute solution of hydrogen peroxide is neutral to litmus but it can blue litmus to red litmus.
Hydrogen peroxide is decomposed into water and oxygen because it is unstable in nature.
Hydrogen peroxide can also act as a reducing agent when other oxidizing agents are present because it can take up an atom of oxygen to give water and oxygen gas.
Now, when hydrogen peroxide reacts with potassium permanganate the potassium from potassium permanganate combines with hydroxide released from hydrogen peroxide and manganese dioxide with potassium hydroxide is formed and the residuals formed are water and oxygen.
Therefore,
$2KMn{O_4} + 3{H_2}O \to 2Mn{O_2} + 3{O_2} + 2KOH + 4{H_2}O$

Note:
Potassium hydroxide is also known as caustic potash and can be used for the manufacturing of potassium carbonate, potassium phosphates, liquid fertilizers and potassium soaps and detergents. It can also be used for some major industrial applications.
Manganese dioxide can be used for making glasses. It can also be used in batteries as the positive electrode of carbon is surrounded by manganese dioxide and carbon in the Leclanche cell.