I want to forward that email thread to my W, but I am sure that is not the right approach.
What do you hope to accomplish by forwarding the email to W? Trust me- I've tried this approach and although it will make you feel good for the next 5 minutes for calling her out, it does nothing but come across as snooping and controlling in W's mind. Besides, it's clear in past conversations you've had with her that she'll just come up with another lie to cover her tracks.
She already knows where you stand on A, so no reason to bring up additional intel. If only I could apply the advice I give to my own sitch