Change of employer and H1B transfer during CAP-GAP......!!!!

[Sri Harsha]

Dear Members,

 

My H1B got approved for the Employer X. with the start date

as Oct 01, 2013 I got a new job offer and i want to move to that before the
start of my H1B i.e Oct 01, 2013.
The new Employer Y is ready to do H1B transfer before Oct 2013

My current work status is OPT/CAP-GAP
My question is
 

1) Is it possible to change the Employer name on my Cap-Gap I-20

    If so, Can I change the Employer name before he apply my H1B Transfer

2) Should I change the Employer name on my Cap-Gap I-20 after the new employee
    apply H1B transfer.
3) Should I change the name of the Employer only after my H1B transfer got approved.

My new Employer/Company Y is concern with the name on the CAP-GAP i-20 and
they want their name over there.
.
Waiting for your reply and thanks in advance.