I’d start by talking to a transmission specialist. There’s a fair chance they will have seen these exact symptoms before in this make and model and will know what’s wrong without wasting any time or money on guesswork. It’s probably also worth going back to basics and checking the transmission fluid level, as low fluid can cause all sorts of drive problems. If, on the other hand, the gearbox is worn out internally, you’ll either need to rebuild or replace it, but don’t rule out something less expensive like an internal selector fault or even a torque converter problem.

This sounds more like a gearbox that isn’t selecting fifth gear properly, rather than one where it’s leaping out of fifth. For it to pop out of gear when cruising along suggests it’s not getting into gear fully in the first place. You might be able to adjust the shifter’s position and alignment and therefore get fifth gear to engage more fully.

There’s also a chance that this refusal to select fifth gear is the result of bent or worn selector forks within the transmission itself, and that’s a gearbox-out job unfortunately. So the first step is to get underneath with a torch while somebody inside attempts to select fifth gear. If the selector linkage runs out of range of motion, you might just have an adjustment problem which can be pretty easily sorted.

Modern cars absolutely hate having low battery voltage. It throws everything into chaos and can easily produce the sort of problems you’re having as the major components of a car are, these days, all controlled by computers.

By turning the car off and on again, you’re effectively performing the old Alt-Ctrl-Del function to reboot a home computer or laptop. As such, and especially since the transmission works perfectly when you reset it, this is almost certainly an electrical problem. A more permanent reset might be needed, so try leaving the battery disconnected overnight and then reconnect it next morning. This will often return things to normal.