I’ve done quite a bit of searching. I’m looking for hook/action called after a theme update. I know the initialization one – but a basic update? Cannot seem to find.
If one does not exist, how can you run a function after auto-updating?
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I have never tried to do what you are trying to do, so no promises, but there is an action hook called upgrader_process_complete in the upgrade method of the Theme_Upgrader class, and it look like it would do what you want. The action is called like:
do_action( 'upgrader_process_complete', $this, array( 'action' => 'update', 'type' => 'theme' ), $theme )
So it looks like that $this parameter should have the information you need.
There is also an upgrader_post_install hook in the parent WP_Upgrader class. That might be useful.