I wonder if there should be a the in front of week in “the end of next week”. Context:
I will give you updates on how things work out by the end of next
The current week began on Monday 19 November. Any reference to the end of the week that begins on Monday 26 November would be to the end of next week. Any reference to any other week would normally be to the end of the following week. I’m not sure that the end of the next week is ever needed, but, as always, I should be interested to hear of counter-examples.