US Citizen Extending Trip?

Wild family drama/dynamics - which is partially hilarious but now can’t focus on work till I find answers haha - brother in law is currently in London - the trip was just a vacation for I think 1 week, so (if I understand newer procedures) he got his ETA to go, and I assume that’s all he has (since just a vacation). He “met someone” within a couple days and thinks “they might be the one”………..supposedly a Brazilian citizen in London that is maybe an architect but can’t find work so is currently working at a hotel?……………ok I digress - he said he is going to extend his trip by 3-4 days because of this person he met.

My random question was, do you “technically a by the book” need to ask for an extension? Or as long as he stays less than 6 months, whatever he said going through customs (oh just here for a week, staying here) doesn’t matter?

We got a chance for a dateline episode maybe within a year but I can’t wait that long for the visa/eta/tourist entry