To add a button to field management you need to modify the below XML file.

administrator/components/com_comprofiler/xmlcb/views/view.com_comprofiler.showfields.xml

We do not support core edits so I can not advise you on how to do this, but that is where it'd be done.

To execute a auto action via URL you need to set the Triggers to None then you can use the URL found on the Global tab of your action.

What exactly are you trying to do and why does it require manually updating a Custom HTML field?

You'd use a Query action with Triggers set to None then you'd add the row id to the URL, which you can then use in your query. You'd want to sanitize the substitution as follows.

[cb:parse function="clean" method="int"][get_row][/cb:parse]

The URL to your action for example would be as follows.

index.php?option=com_comprofiler&view=pluginclass&plugin=cbautoactions&action=action&actions=ACTION_ID&row=ROW_ID_HERE

In your Custom HTML field or wherever you need it you'd supply a link to the above URL usage to directly execute your action.

To redirect back to profile you could create Redirect action that acts on autoactions_onAction where you'd then condition against [var2_id] to ensure it's your query action.

In short it is possible for CB Auto Actions to provide standard component endpoints. So in short you could theoretically make an entire component just chaining actions, but that's a pretty advanced usage.