Hey there @Bravo4u - thanks for posting and welcome to our community! 

To clarify - you are the invitee in this situation and not the host in Calendly, correct? Just to be sure! In that case, this is not something that will depend on your own Calendly account at all nor is it something we can help with re: Calendly settings on your end. 

When you book with someone else via Calendly, that booking will not be communicated to your Calendly account or appear there. 

If you are using Google Calendar and are accepting meeting invitations but those meetings aren’t appearing on your Google Calendar, there is likely something going on with either your calendar, browser or device settings.

Try the following: 

1. clear cache and cookies (steps below!) and restart your browser
2. check your browser for application updates, run any that exist, restart your browser
3. check your device for software updates, run any that exist, restart your device
4. try an Incognito window (Chrome) or private/secret window (other browsers)
5. if possible - try another device entirely to see if it works - then you’ll know if it’s a device issue
6. restart your browser and/or device after each step as needed
7. if all of this fails - reach out to Google Calendar community here or Google Calendar support here for more assistance

How to clear browser cache

I hope this helps!  

To be clear, I am merely the recipient of calendar invites from Calendly, from multiple different people. When I get the invite, I click “yes” but get this email response: “Sorry, Google wasn't able to send your request to <sender’s domain>”

It also doesn’t add it to my calendar. My workaround is to import the .ics. 

This used to work for me. I am using legacy Gsuite.