Marrying a LPR

[CIA]

guys and girls;

is it possible to marry an LPR and then file AOS the date the LPR moves to have a certificate of citzenship.

My guy friend wants to marry his girlfriend, he is a legal alien, she is a UK citizen, shes going there on VWP this year, he is filing for certificate of citzenship right now,

IF they were to get married on her VWP can they file AOS when he gets his citizenship, which means the girl will be overstaying her VWP.
They can't file for AOS after they get married because the guy has no supporting documents (i.e proof of citizenship) his parents are USC but he is not (another long confusing story)!

thanks!!!!