At the end of Book 2, Azula zaps Aang with lightning, apparently killing him. We know he survived thanks to Katara's healing. Azula gives Zuko the credit for killing the Avatar, somehow aware he has survived. I get that Zuko suspected it, because he knew Katara had the water from the koi pond at the north pole, and I get that Azula could have noticed Zuko's doubts, but what if his doubts were unfounded? What if the healing didn't work? If Aang really had died, Azula would have just given Zuko credit for personally defeating the greatest bender on earth and the main threat to the Fire Nation. To me, this seems like a huge risk, she's betting everything on a teenager's understanding of the spirit world. It seems like she must have more evidence than just Zuko's doubts that would make her more sure. So what is that evidence?
It may not be as big a risk as you think. It's likely that she doesn't know for certain whether Aang is actually dead, but by pinning the deed to Zuko, she gets benefits either way:
- If it turns out Aang is not dead, as she begins to suspect, then Zuko takes the blame for the supposed incompetence and will be blamed for anything that results. Discovering the Avatar is not actually dead would almost certainly result in Zuko being dishonored and exiled again, and it would remove him from the line of secession to the title of Fire Lord (to which Zuko is first and Azula is second). End Result: She has successfully maneuvered her major rival for power out of position, and she becomes Fire Lord by default when Ozai passes it down.
- If Aang is actually dead, then by allowing Zuko to take credit for it, she has effectively blackmailed him into loyalty. And even if Zuko does eventually attain the title of Fire Lord, she would presume to have the skill, in both charisma and martial arts, to scheme against him and take the title from him at a later time. Aang's actual death will also greatly weaken any opposition to the Fire Nation. End Result: The Fire Nation becomes a world power, largely without her continued assistance. She then has blackmail over the next Fire Lord (Zuko) and believes she can otherwise take the throne from him at a later time (easier than she would trying to take it from Ozai, anyways).
TVTropes calls this sort of planning a Xanatos Gambit - a maneuver by someone where any result still moves their plans forwards, and this exact scene is provided as one such example (under Western Animation) with some of the points above.
Azula already has the prestige and respect so killing the Avatar won't come as that big of an advantage for her. She also promised that "he can have whatever he wanted" if Zuko switch sides. If Azula gave more credit to Zuko then this would ensure the restoration of Zuko's honour and reputation rather than him being a side character in the siege of Ba Sing Se.