This seems like a firmware issue to me.
I got my Osmo Action yesterday and have been doing some tests with the Mic Adapter while  hooked up to external power and the the screen does not shut off according to what I have it set to in the settings, works fine when just using the internal battery.
Just because it's on external power does not mean the screen should stay on the whole time.

If I am doing some setting wrong let me know :-)  

Thanks :-)

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