E.F.
M.,
My kids have done this too, What I have ended up doing is giving an early a nap as they will take and wake them after 45-60 min. I will say, it is hard to wake them, but if I do not they will stay awake until 10-11. So I shoot for like 2:30-3:00 nap and get them up at 4:00 or before. I try to have something fun to help wake them up, like some place to go or a fun snack or activity. This helps a lot. but there are days that I repeatedly have to wake them and wake them and wake them. After about the 2nd time I will just bring them out to the couch and "life" keeps them awake pretty well.
Good luck finding something that works:)
E.