Automate Button Press on external device

I’ve got a 3G dongle stripped back shown below. The Large circle on the front picture is the on off button. By holding that in for 5+ seconds the device turns on and off. I want to run this off a relay. Any ideas how I can bypass the switch or replicate the button press with a signal?

It depends on your skill at desoldering the button. If you think you can do that and hook wires to the remaining pads then go right ahead with a relay or optocoupler.

If not, then a more mechanical solution with a small hobby servo might be better.