grep – How do I transfer certain parts of the file to another file?

I have a FILE1

NAME START

Hi, how are you?
Nice to see you

begin
Aaron
Kyle
Robert
stop

Only for official use
Stamps here

begin
Riley
Gayle
stop

Bicycles here
United farmers

begin
Alex
ford
Sergio
stop

STOP NAMES

I would like to do the following:

If the string "NAMES START" exists, transfer the contents of each "start" and "stop" into a new FILE2, omitting the start, and stop in the new FILE2.

FILE2 should look like this:

Aaron
Kyle
Robert
Riley
Gayle
Alex
ford
Sergio

Help please!