Groupjive edit group doesn't allow change parent

Hi, I've been trying this moderator thing. The problem is that I have one user that is moderator (in groupjive) of several groups, but he can't edit, it's like a normal user.

A Group Moderator is not a CB Moderator. A Group Moderator can not edit a group. They are there to moderate content of that group (publish, unpublish, approve, etc..).

So a GROUP MODERATOR can not edit groups, but GROUP OWNER/ADMIN neither? Only joomla admin can do this? Or can I add the user to a special user group?

Where I can make a user CB Moderator?

Group Owner can freely edit the group. So can the owner of the Category that the Group is in.

Group Admin can not edit the group, but they can edit various content in the group and do Group User promotions. They're a step above Group Moderator, which is there to strictly monitor content.

Where I can make a user CB Moderator?

It's based off usergroups. So whatever you set the moderator group to within CB > Configuration > Moderation is what you need to give to your user to make them a CB moderator. This is all detailed within your CB documentation, which you can download from the Advanced downloads section; it will explain all the basics of CB it self.