I am usually one to suggest an evaluation, but I think your clue is that she is fine at school, and she probably would not be if she were going to have a developmental, neurobiological or neurolgical behavioral issue. They can't turn it on and off. She sounds like a handful right now, but you want to have the behavioral issues at home, and not school, not the other way around-which can mean more serious issues for you later. Right now, I would try reading some books like How to talk so your children will listen, and just be firm with her.
On the other hand, if she is responding well to the structure of the classroom, you may want to take a look at why that enviorment is producing sucess for her. If they are using positive interventions (rewards for success that she can see) this may be more effective for you than punishment is. I would suggest firmness, but an adjustment in your approach. Every kid has a currency, and at somepoint, when they are not sucessful at all, they quit trying, this may be where she is with you. If you take all the fun things away, then give her small items back for the positive behavior, maybe she can do that. My point is, losing too much, too fast, as a punishment, may just seem hopeless if she is not having any sucess at all. Sucess breeds sucess, and she will want more of it. Something that may work for you too, is to tell her what you want her to do, in small chunks, instead of what you want her to stop doing. If she hears exactly what you want for the next five minutes, and she can comply, reward her.
A token econemy works sometimes for kids this age. They get a poker chip every time you catch them doing the target behavior, then you give them a list of the behaviors you want, and for the most difficutlt things, like lasting all day without being nasty, she gets a handful, when she says please, self corrects her attitude, or complies when she would rather say no, she gets one chip. She can buy stuff with the chips, from the most basic TV time, to the toys you have put away, to going to the library, or renting a movie, or she can save them for something bigger, what ever fits your lifestyle and her likes/dislikes. The point is, you take it all away, she earns it all back, but it is kind of easy for her to be successful at the little things, harder for the medium things, but a bigger pay off, and really hard for the biggest things, but the pay off is really, really good, and she can feel and see how her success pays off. Sometimes this is much, much better for a kid this age than a sticker chart. But the most beneficial thing about it is that it is a positve renforcement system, she can see success instead of brooding about what she has already done wrong.
Last, if you see a vast difference between the enviornement at school, meaning that the structure is set up for her sucess, or she is having issues at school that have not been reported to you that are just not yet causing a concern (impulsivity, playing alone, social rejection, fine motor issues, frustration, etc) then add up what you have and seek out a Developmental Evaluation. What I tell people is, if the issue is gloabal, and if it interferes with her life, then it is time to seek help.
It would be best to see a Developmental Pediatrician. They do full evaluations that will tell you if you have a child with a developmental issue, or just one that is difficult.
Another option is play therapy or family therapy. Espeically if the issues are just between you and her at home, and it is not a global issue, and if revamping your dicipline plan does not help you, you might benefit from some outside help, someone who can tell you directly, based on observations, if you have more issues than just mother daughter friction. It can be very short term help. They will also be able to help you decide if you need to seek more help.
Good luck. 4 is difficult!
M.