How to put users to the user groups

The description on the plugin page on this site states fairly clearly what the plugin and its options do:
https://stephensherrardplugins.com/plugins/pta-volunteer-sign-up-sheet-groups/

Perhaps there was a misunderstanding on what the groups plugin does? It allows you to create groups of sign-up sheets, which can be thought of more like categories. Then you can split them out to different pages on your site by using the group shortcode argument.

My plugin does NOT create any type of user groups. So, if you want to limit certain users to only be able to view certain sign-up sheets, what you need to do is use some other type of content restriction plugin to create user groups and then make it so that only certain user groups can view certain pages… then you can assign your sign-up sheets to their own groups and use the appropriate groups shortcode on the pages to show the group sign-up sheets to the users that have access to that page.

One such plugin you can use to create user groups is the very excellent, and free, Groups plugin here:
https://wordpress.org/plugins/groups/

Other users have done exactly what is sounds like you are trying to do by using the combination of the WordPress Groups plugin and the sign-up sheets groups plugin, setting things up as described above.