This is what I did for hitting (I am a former preschool teacher, so if this all seems a little 'coachy', bear with me)
First, I listen to *both* children and take evidence into account. For example, if Sonia's face has a red mark and Todd says he 'didn't do it' and Sonia said he did, I'm going to assume Sonia's telling the truth.
I help the person who was hurt to express themselves. "I don't want you to hit me. It was okay for me to go away, I was done playing Barbies. When you hit me, I don't want to play with you." or whatever they need to say.
(I have the child who hit/hurt, go with me to get an ice pack/bandage/first aid for the hurt child. I would do this sooner if there was blood or urgent need for first aid, of course.)
I do try to talk the children through making amends/apologizing, however, I never force an apology. IF the child who hit is sorry, and obviously contrite, I may let them resume playing. But this needs to be a genuine apology, otherwise, then the child who hit may be told to go sit alone or take a break, usually out of the way of the action going on. In my preschool, I had a 'cozy corner' ( a quiet place for kids to sit out of sight of the other kids and pull themselves together) ; outside, the person who was hurting/hitting had to sit on a step alone.
Last, I check in with the child after I see that they have cooled off. If they haven't apologized, I do ask if they would like to 'try to fix things with your friend' and if yes, they can go back to where they were playing. If no, then I would direct them to a different activity.
In your friend's situation, I do wonder if some of the hitting isn't an attention-getting behavior. In the situations where there is repeated hitting going on from one child, I would have them stay with me throughout their time at preschool or in my care. This actually proves to be very useful for many children. "I see that you are not being safe with your friends, so I need you to be with me now." And then, boy, do I get boring. This is NOT a time to play with the child. They must keep themselves entertained with whatever is at hand, but they may not go out of my sight. "I'm not here to have a cheery conversation with you. You are here because you weren't being safe with your friends. I have work to do now." Wash dishes while the child works/plays at the table. Fold laundry. No special attention, but they are getting the drift that they don't have any liberty. To most kids, this is far more instructive than just sending them to their rooms.
When the child has spent enough time for the parent to feel ready to release them (they are calm, body relaxed, etc.), then we make a plan for the future. Sample script:
Parent: "What happened so that you needed to be with me?" (I never ask 'why', as it is often hard for kids to articulate that one)
Child: "I hit Suzy."
Parent: "Right. What was the problem you were having with Suzy."
Child: "She stopped playing Barbies and I wanted her to keep playing."
Parent: "And that didn't work for you or Suzy. So, the next time a friend is done playing, what can you do?"
Child: "I can keep playing my thing. Or I can do something else."
Parent: "That sounds good. The next time your friend is finished playing, you can keep on with what you like, or you can find something else to do."
It's important to also insert a little empathy into the situation when it's not a time of conflict. Your friend can talk about "sometimes, it's really hard when I want to do something and no one wants to join me. Do you ever feel like that?" This opens up a safe way to talk about negative feelings without validating the misbehavior in the moment. The more children feel that we understand *where* they are coming from, the more likely they are to comply with our requests for them to regulate themselves, because they feel we are an ally who has their best interests at heart. "I don't want you to get into trouble for hitting again, I know you don't like sitting out. I'm trying to help you figure out what would feel better for you when your friend walks away from what you are playing." The book "How to Talk So Kids Will Listen...And How To Listen So Kids Will Talk" by Faber and Mazlish is also an extremely useful book for parent/child communication (actually, it's excellent for most interpersonal communications). I highly recommend both you and your friend giving that a read.
Ultimately, it's not necessarily about 'what kind of punishment' but really "how do I help my kid buy into changing a behavior so she knows it's to her benefit?"