"I might do it" - The event of doing it is in the future, and you are still considering if you will do it or not. "I may have done it" The event of doing it is in the past, and you cannot remember if you did it or not. "I could do it"- You have the ability to do it.
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Fandorin"I might have done that " sounds strange since it is used while expressing action when we don't know whether or not it took place.It's not strange - often used when we know something happened and are speculating who did it or how it happened.