As Zmrzlina says, it can't mean that. To leave something behind is sometimes att lämna något bakom sig, but that is more about events in life. There's another expression lämna efter sig which refers to leaving something as a legacy, typically after you're dead. If you leave a key behind, we would choose between expressions like lämnar kvar or glömde 'forgot' or fick inte med sig ≈ 'didn't manage to bring along' or something like that.
No, this sentence means giving the key to a person, either directly (handing it to them) or indirectly (putting it somewhere where you know they will find it). Putting something back simply means you return something to the place where it belongs, without intending for any person in particular to find it. That translates to "lade tillbaka" ("lägga tillbaka" in infinitive) in Swedish.