See also the official documentation:
Help Center > Members Manual > Working with calendars > Subscribing to a group's calendar

How To Sync Your Google Calendar

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).

Displaying calendar events on your own website

If you wish to use Agenda view on a website in order to display as many upcoming events as possible in overlapping months. How many weeks/months are displayed depends on how many events are packed into a week. The Agenda display allows endless scrolling in devices instead of having to click through each month. 

First get your GIO calendar set up into Google Calendar. Next choose "Agenda" as your display of choice. Then go into settings and configure your display size if desired and get your embed code. 

The embed code will default to an Agenda display listing on your website. It also allows visiting viewers to click through the tabs to display in monthly and weekly views if they so prefer. In monthly views the visitor can view and print up to 6 months. 

How to subscribe with other calendaring software;
how to print calendar

Subscribing your calendar to Outlook or Google calendar or some such will allow you to take advantage of the more versatile view/print options found on those platforms.

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.