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As of this writing (20MAR2018) you can sync your Google calendar with the groups.io calendaring, but not the other way around. A full integration with Google Calendaring has been requested in beta.

At the bottom of the Groups.io calendar page is a blue "Subscribe to Calendar" button. Click that and copy the URL it gives you. Then in the Google calendar click "Add a friend's calendar" and select "From URL". Current evidence suggests that Google will refresh by reading the URL once a day.

You can set the calendar to "Public", even if your group's Messages are private. Which means that you can put a link to it on the website, or with some greater effort maybe use an iFrame to embed it in the other website (not yet attempted to my knowledge).

The official Groups.io user documentation is in the Groups.io Help Center.