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This question already has an answer here:

I have been able to set up an automatic forward of all incoming Gmail, but for outgoing I set up a filter, and it doesn't do anything. I need the forwarding to be automatic and discreet.

I am going to try the Gmail Auto-forward script.

marked as duplicate by Rubén, ale, Alex, codingbadger Jul 29 '15 at 6:54

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You can use the Gmail Auto-forward script bound to a Google Sheet. In Gmail, you create a filter that marks all matching emails with a filter. On the sheet side, specify that label name and the forward address and it will send them automatically.

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