What is the problem with my GFCI outlet?

[Bill Howatt]

I'll go along with more information required but this GFCI works just fine for periods of time when the devices are connected (normal circuit use) and turned on or not turned on. It also is happy with disconnecting the power by switching the breaker on and off and it only fails intermittently when the power is restored after a failure. If we are dealing with a problematic device on the circuit, I'd expect it to be failing frequently regardless of power failures.
The one device I would check out, mentioned by Jim Koepke, is a surge protector since it could do it's protection operation because of a large spike during a grid restore and this could drain some current to ground in the surge protector causing the GFCI to trip. A slightly complicating issue is that the surge protection may be inside some electrical equipment like a TV power supply, not just in obvious plug-in and power-bar protectors. So it would be prudent to disconnect everything if troubleshooting.

[Jim Koepke]

So it would be prudent to disconnect everything if troubleshooting.

One of the difficulties is this only happens (from the original post) when there is a power outage (for a few seconds) and happens about half the time.

A lot depends on the circumstances and could be related to how the power outage occurs. Some power companies have their own automatic circuit breakers. These can open and close during overloads when weather causes lines to cross or trees to touch a line. There could be two different line breakers in this circuit where one opens and closes cleanly and the other causes surges.

Like said before, there could be a lot of causes. This includes a GFCI outlet being more sensitive than others.

jtk