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No, Order Manager will not send automated emails to buyers when you mark orders as Cancelled, Fraudulent, or Returned.
There are three instances where a shopper will receive an automated email from your store:
Your customers will see an order status update for cancelled, fraudulent and returned orders on the Order Confirmation page itself; but they will not be emailed or notified to view the Order Confirmation page for a status change.
Shoppers will see the following messages either in or above the Confirmation section of the page depending on which version of checkout the merchant is using and how the order is marked:
Your customers will not see information added to the Notes section by you, the merchant, or by Turbify. This field is labeled Merchant Notes and it is for merchant reference only.
Merchants who wish to email buyers can use the info from the Email field in the Order Manager to send a message. If you have an email client set up, you can click the email address link in the order to send an email. If you use business email, you can copy the email address, go to your business email account, and then send an email to the customer explaining the order status if you wish.