Keep Outlook Invitations in Inbox

Yes, it's a configurable option in Outlook.

Outlook 2003 and 2007

1. Go to:

   • Tools > Options > Advanced E-mail Options (Outlook 2003)
   • Tools > Options > E-mail Options > Advanced E-mail Options (Outlook 2007)

2. Untick Delete meeting request from Inbox when responding.

   

Outlook 2010, 2013 and 2016

1. Go to the File tab in the ribbon and click Options. Select the Mail entry in the left pane.

2. Scroll down and untick Delete Meeting Requests and Notifications from Inbox after Responding.

   

Now when you accept a meeting invitation it should keep the original mail in the inbox.

On Outlook online, go to Options -> All Options -> Settings -> Calendar, then uncheck "Delete meeting requests and responses that have been updated" then press save.