The Steam group permissions suck. The only way to allow everyone to invite others by default is to have the group be open.

Instead, everyone currently in the group has been made a moderator to grant them invite / request permissions.

Currently, moderator permissions enables you to:
-Create & edit announcements
-Create & edit events
-Choose a player of the week
-Review games for the group
-Broadcast to the group's curator page
-Set the group chat privacy
-Kick users from the group chat
-Invite people to the group chat
-Add/Remove comments
-Invite and manage application requests

This will remain in place until a better solution becomes available. Enjoy.