Hi and welcome to the BW

,
I think you may need to tweak your routine a little, at 5 months, try and aim for 3 naps as she is taking too many naps and not all of them are restorative. Her bedtime is also quiet late (if she's waking up at 5am for the day).
I would probably start with the last nap/bedtime. Rather than giving her the 4th nap, put her down for the night around 2-2.5 hours from the time she takes her 3rd nap. Your day is quiet long and she is probably quiet overtired from not getting enough sleep at night. Its common for overtired babies to wake periodically at night and then be up for the day at a very early hour. When she wakes at 5am, can you do a quick feed and put her back to sleep or she awake and can't get back to sleep? If she does go back to sleep, then that would be a good way to start but I would suggest waking her up at 7am or if she wakes earlier, from 6-7am, that's fine too.
Then, I would encourage you to try and push her morning A time so that her morning nap is closer to 9am and I would not let her sleep more than 1.5hrs. Ideally you want her to take 2/1.5hr naps and a catnap and sleep around 11hrs at night. If she takes a long morning nap, the afternoon nap is likely to only be short. Also, a long morning nap is likely to reinforce the early wake ups.
With regards to A times - see if you can stretch her awake times to at least 2 hrs.
Are you feeding her at night at all? Would you say she's an independent sleeper? How do you resettle her when she wakes at night?