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).