Participants in a group conversation have different roles. You are either a group admin or a group member. The roles are synced across all your devices and on all platforms. Every group conversation has at least one group admin who holds the highest level of permissions.
The creator of a group is by default the group admin and is the only one who can also delete the group conversation. There can be several admins.
See also this video.
Group Members
Group members have no permission to manage group conversation settings, but their personal ones.
Permissions:
- Notification settings
- Archive a conversation
- Timed messages
- Leave a group
Group Admins 
Group Admins manage the roles, the conversation´s name, and settings.
Permissions:
- Add and remove members
- Add and remove services
- Turn allow guests on or off
- Turn the global timer on or off
- Upgrade and downgrade participants
- Rename the group conversation
- Enable and disable read receipts
- Group members´ permissions
As a group admin upgrade and downgrade participants' roles:
In the conversation list:
- Select the name of the group:
- Select the group name at the top or the info icon:
- Select the participant:
- Toggle Group Admin to on or off: