So, one of the last swimming tidbits Igor shared with us (which was pretty novel to me) was: in the fly, don’t let your arms follow your head down or shoot them deeper/downwards as you would in breast, but keep your arms slightly above the head. You’re still leading with the head in terms of making the undulating motion, and as Igor used to say “pulling with your head”. And, you’re still making the second kick stronger than the first to “kick your arms forward”. But, by not letting your arms follow your head down, your second kick becomes a lot better and isn’t unnecessarily huge.
Since that was not that eloquent of an explanation, here’s an underwater video of Phelps doing it, which demonstrates it a lot better!