Originally Posted by MKE-MR
I would approach this as two different problems.
1. Check the Minimum Connecting Time for LAX intl to domestic. I think 52 minutes is doable as you will arrive and depart from the same terminal area (6&7). If you were coming in at the Tom Bradley Terminal I'd say forget it, but 52 minutes isn't disastrous (unless something goes wrong of course).
True, they will be arriving and departing from the same terminal. But once they clear immigration and customs they'll have to exit terminal 6 and go upstairs to terminal 7 and clear security. Depending on the time of day there can be significant lines, even in the elite TSA queue.
With all carry on luggage I've done it in 30 minutes, but would consider that the exception. With checked baggage you probably wouldn't make it.