The correct option is A taking
Let’s examine the sentence: “She finally admitted ____ the key.” It states the fact of taking rather than the action of taking. The ‘fact of taking’ is a noun or gerund and the action of taking is verb. The gerund always has the same function as a noun (although it looks like a verb). The gerund ends in ‘ing.’ This rules out Options B, C and D. So, Option A is the correct answer.