Groundwork is more than just lungeing. Lungeing is great for getting a horse warmed up, introducing new tack, etc. but usually when I say someone needs to do more groundwork, I'm thinking of exercises that build respect. Anything from teaching a horse to lead (staying with you on a loose rope, stopping when you stop, going faster when you speed up, etc.) to trick training will, when done right, teach the horse to respect you.
Your horse coming in when you try to lunge isn't a problem of not knowing how to lunge- as you said, she lunges just fine with your trainer. It's a problem of your horse not respecting your leadership, not believing that you will follow through with your command, or your timing/technique is off, etc. It's not something that most people are good at right away and takes a lot of practice. Unless your technique is way off, doing other groundwork exercises that build trust/respect will make your lungeing better, too.
Ask your trainer for help, and if necessary, record your sessions alone with your horse to show her. I know that personally, my horse acts WAY better when a trainer is around, even if she's just watching. Last time I had a lesson with a NH trainer I said my horse is always lazy on the lunge line and he made a total liar out of me