Hi @Stig Kølbæk,
To the best of my knowledge, it's not possible using the built-in capabilities of either Exchange Online or Outlook client.
Since you are looking for a rule solution, resending it manually as mentioned in the article below is not an option for you, right?
How to forward without header info in message body in Outlook?
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Regarding the possibilities involving plugin or script, actually I'm not an expert when it comes to development, but I tried searching a lot and so far cannot find any direct solutions for the specific requirement either. I do find some VBA scripts to send a new message using the body of the received message, but the rule just wouldn't fire at all when I tested it in my Exchange 2019 lab environment. I'll leave the links I referenced below in case you are familiar with scripts or have the interest to take a further look at it:
Then please understand that the tags "Microsoft Exchange Online" and "Outlook" are focus on general questions related to Exchange Online or Outlook, if you would like to dig deeper into the script solution, personally I'd recommended posting your question using tag "Office Development" or "Exchange Server Develpment" instead as community users there could be more familiar with scripts. Thanks for your understanding.
Meanwhile, I just tried posting the idea into the dedicated feedback portal for Exchange Online. I'll leave the link below in case you or other community users would like to add votes or comment there as well. Hopefully this feature can be added soon.
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