While a group has permissions controlling who can send, an organization owner or group administrator can override those permissions for a particular group member.
A group administrator or organization owner can go to the member list for a group, select a user and adjust their permissions.
Scroll down the list to the sending rights and choose whether their messages:
An organization owner can go to the organization member list, select a user, choose the groups to control. Then by going to edit settings and selecting sending rights, choose whether this user's messages for the chosen groups: