While "mission" might seem like a close relation, it's not the root word. "Commission" comes from the Old French word "comission," which is formed by combining the prefix "com-" (meaning "with" or "together") and "mission." "Mission" itself comes from the Latin "missio," meaning "a sending."
So, the root word "commit" is the basis of "commission" because it implies entrusting someone with a specific task or duty, which is the core meaning of a commission.