I feel bad for both kids involved, six is a very young age and I think kids need to learn, but also be treated with compassion since they are just starting to navigate socially.
If I were you, I definitely would let the mom know that you appreciate her bringing it to your attention and that you are working on it. I also think you should let the mom know that your daughter is going through a lot since you've gone back to work and are working through a divorce and perhaps this is contributing to the behavior.
I have mentioned before, we always used playacting to work through social situations. You can start with dolls (we used Barbies) where your daughter is one Barbie and you are her friend. Start easy, and have your daughter's Barbie try to play with her friend. You say something like her friend might say, and your daughter needs to practice several responses. Once she's comfortable with the Barbie dolls, you can just playact without them and she can be herself and you are the friend. This has ALWAYS helped my daughter come up with different scenarios and solutions. We talk about how her actions and words are affecting the other person. We also will switch, where SHE is the friend and YOU are her just so she can see what it feels like to be the other person.
Keep it light and fun, and try to make her come up with the solutions first. You can guide her, but it works best if she is working through the problem herself.
You can also use this method to help her get over her shyness in going up to other kids. I agree, see if you can create playdates with other kids in her class so she can make new friends as well.
Kids are not born knowing how to be social. It's something that is learned through trial and error.
My daughter was the bossy kid on the playground and she had to learn that no one wants to play with someone who is bossy. When she had no friends, she would pout and cry. We worked through it for sure, but it was a tough time.
Good luck! You'll be helping her navigate social situations for a long time to come!