Hi @Roger Roger ,
1.If we want to check the attachments contained in the mail, and according to your needs. Creating transport rules will be a good choice.
2.You have a remote shared mailbox, so we should create a transport rule in Exchange online Admin center.
3.According to the inbox rules you created, you want to keep the email in the shared mailbox and forward to external recipients, right? If so, you can create a transport rule as shown in the screenshot below. The rule is that if an email is sent to the shared mailbox "SharedRe" and the attachment contains files in the format of ".jpg" and ".pdf", the address of the contact you created will be added to the email "Bcc", and the email will also be sent to the contact.
The transport rule created above will directly redirect the mail during the mail transmission process without modifying any attributes of the mail itself. However, in emails received by external recipients, the sender will be shown as the sender sent to your shared mailbox instead of the shared mailbox. For forwarded mail, do you need to satisfy the sender to be displayed as "Shared mailbox"? If necessary, because Inbox rule cannot recognize the attachment, you need to follow your way. Make some parameter changes to the mail through the transport rule, and then forward the mail through the parameters in the inbox rule. But it should be noted that the object of "Forward to" in the example you provided should be an external contact.
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