A calendar can be made public to allow others to see and sign up for events, which will also automatically add them to the group. 

To make a calendar public, choose Calendar settings in the upper right corner of the calendar tab.

At the top of the Calendar settings, choose "Allow anybody with the link to access the calendar." 

Once published, the URL for the public calendar will appear at the top of the Calendar page. Copy the URL using the clipboard icon to email or post to your website.