question

Hi @PaulSwanson-5602,

What's the version of your Outlook for Mac 2016(Outlook>About Outlook)? please ensure that you have updated to the latest version.
Is the issue only recently occurred? If so, have you performed any specific operations before it?

I can see the email in Gmail. I can see the acceptance in Outlook Calendar. However, there is no email visible in Outlook Mail.

According to your description, it could work normally in the Web Mail and other client, so the issue may be related to your Outlook for Mac itself.

As I know, client data is synchronized from the server, the sync error between client and server may cause this issue, so it's suggested to remove your account and re-add it on your Outlook(Tools>Account>click"-"to remove your account and click"+" to re-add your account) to test if the issue has any difference.

If the issue still exists,corrupted profile may cause it, please try to create a new profile and re-add your account (Finder>Application>Outlook>right-click Outlook>show package contents>content>SharedSupport>Outlook Profile Manager>click"+">set as default)to check if it's effective.

Update 1:
If your account could work normally on other machines, the issue may be related to your corrupted Office database or Outlook client, it's suggested to rebuild your Office database or completely uninstall and reinstall your Office for Mac.

If it also couldn't work normally on other machines, according to my research and tests, my Gmail account could receive the meeting response on Outlook for Mac and I haven't found any settings in the account and Outlook that may cause the client hide these messages. I would continue to research this issue and if I have any update I would post back here.

Update 2:

For some reason Outlook is simply not recognizing these as emails. When I tested with Mac Mail, the messages appear fine.

As I know, there’s no setting in Outlook client that could cause this issue. I have indeed tested a lot with my personal Gmail account on the Outlook for Mac 2019(version 16.42) , and it could display the response normally like below whatever the respond is from web mail or Outlook. So if it’s possible, please try to update your Outlook client to the same version with me to check if it could work normally.

Also, considering there may be any differences between personal and organization accounts, to further analyze your issue, it’s suggested to add other personal Gmail accounts on your Outlook for Mac to test if it could receive the response normally. If personal Gmail account could work normally, the issue may be related to your accounts themselves in your organization, please contact with your mailbox administrator to get more information.

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@JadeLiang-MSFT - Thanks for the response, but I don't think that solved the problem.

You asked about the version of Outlook for Mac - I am running version 16.33.
You suggested that the problem may be a corrupted account or profile.

I tried updating Outlook for Mac and creating a new profile, but that did not solve the problem. See this video: https://www.loom.com/share/dd5883b44cf14bd680edeabf849aa014

This problem seems to be common on multiple machines and is happening very consistently. For some reason Outlook for Mac is not seeing email responses as actual emails. This is a real problem because it makes managing calendar invitations, acceptances, etc. impossible with Outlook.

Please help!

I am having the same exact issue using Outlook for MAC v16.47. All other clients whether iPhone Mail, iPhone Gmail, emClient on Mac see the acceptance email fine.