How To Sync Your Google Calendar
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).
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You can subscribe to the groups.io calendar using your favorite calendaring software and use the print function in that.
At the bottom of your groups.io calendar is a Subscribe to Calendar button.
Click that and a dialog will pop up with a URL that can be used to subscribe your local calendar to your groups.io calendar (the URL ends in feed.ics).
An ICS file is a calendar file saved in a universal calendar format used by several desktop and web calendaring programs, including Microsoft Outlook, Google Calendar, and Apple Calendar.
Once you complete this subscription and the calendar entries have synced (can take a few hours) you can then print your calendar using the external app. The two calendars will continue to sync at regular intervals and you don't have to do it every time.
The official Groups.io user documentation is in the Groups.io Help Center.