By default, WordPress doesn’t let you save HTML in the tag description field because a malicious user could post code that compromises your site’s security. Some plugins, however, remove this restriction and let you use images and formatting in the tag description.
The Tag Groups plugin allows HTML only if the user has the unfiltered_html permission. By default, only administrators (super-admins on multisite) have this permission but you can use a plugin to create new user roles and change user permissions in order to grant the unfiltered_html permission to all users you trust.
If you are sure that you can trust all users of your site, you can also activate “Allow HTML in tag description” in the “Front End” settings, under the “Themes and Appearance” tab.