Sometimes with kids, in addition to doing what the Dad seems to be doing per your description... a child NEEDS to be told, directly with no pussy footing around "THAT IS WRONG. Stop it now."
Kids, often are not actually told that something is "wrong" or "right" or "good" or "bad." But rather, they are explained to about things. And after awhile, kids just tune you out. Especially when explanations are too wordy or long winded.
A child need to be told and reminded "NO. That is WRONG." And "correct that... redo that." And then, the child has to learn problem-solving.... to do so. Which is a learned... skill. Have the child, come up with CORRECT, reactions. Which you can offer ideas, but the child has to learn, how to come UP with, alternatives.
It is taught.
Not an inherent skill.
Role play, with her.
And when she is being mean/aggressive, put her in time out.
And TELL HER "That is mean. That is wrong. YOU KNOW THAT." A kid also needs to be told, that the parent is onto them. And that they cannot be manipulated.
I have actually told my kids "I KNOW YOU are trying to manipulate me. It will NOT work. Redo that... and when you do, you come to me... and tell me. SHOW me." And then I walk away.
Sometimes, their "punishment" is that I take myself... away from them. ie: I walk away. I do whatever I am doing and continue to do so. My kids then see and realize, that I mean it. That I cannot be manipulated, and that, I will not put up with it. And that... THEY are responsible, for CORRECTING their poor behavior, TOO. Because, even if punished/scolded.... some kids don't care. They will endure the punishment but still be, icky. SO, you need to make the child, RESPONSIBLE... for coming up with ways to CORRECT, their bad behavior. Kids need to learn, these skills. And be told "NO. That is WRONG. YOU KNOW THAT...."
My son is 6. My daughter is 10. Often as parents you just assume a kid knows right from wrong, already. But, I have found, that with my son... that when I actually tell him "THAT IS WRONG. Do not do that..." it jars his conscious. And it makes him realize, again even if he is 6.... that his behavior is wrong. And that, no one has to put up with it. And that I won't.
The books "Have A New Kid By Friday" and "How To Talk So Kids Will Listen, and Listen So Kids Will Talk" is good. Good ideas in it.
And again, with young kids, make sure they get enough sleep. Adequate sleep. And that they are eating enough and well. Many young kids, need to have snacks in between meals. Otherwise, their blood sugar levels drop.... and this is when they get moody or fussy or cranky and their tolerances, drops. And when overtired, too.
A child need to be told, if/when they are being mean. And told that it is not acceptable.
Don't pussy foot around it.
The girl is in school?
How do the teachers deal with her?
Is she like this in school too?
You need to show her LIMITS.
Boundaries.
When she is like this... tell her "tantrum and yell all you want. I am leaving. I have things to do. When you are done, you come and tell me..." and walk away.
A kid, WILL burn themselves out.
They will, deflate.
But you have to tune them out, while doing so.
And once they deflate, THEN you tell them in no uncertain terms, that you will not put up with it, it is wrong to act that way, and they need to APOLOGIZE to you.