This is now possible with Google Music Manager. Here's a tutorial on how to use Google Music Manager. For more information on Justin's findings please continue reading.

Today November, 17, 2011, I spent some time trying to figure out how to sync music back to iTunes that you "pin" to your phone in the new Google Music app. I have nothing against cloud streaming services, but iTunes has tons of options that I'd like to keep using. As such, I want to get the music I purchased out of Google Music and into iTunes. I doubt I'll be the only person looking to do so.

After some research, here is what I found.

The files from Google Music are stored with a ".nomedia" file.  This file tells the Android OS, "No other music player would be interested in this".  It was this way through the beta, but I'm very surprised that Google is dead set on keeping music you purchased locked into their music app.  They know it is far from the best ones available and I thought the motto went something like "Don't be Evil".  Weren't they in a tussle with Facebook because you could get your info out of it easily?

Moving on from the ".nomedia" file.  We can also see the files are named "1201.mp3".  I thought, OK, they are just making sure if you have two songs with the same name, they won't have any conflicts, right?  Maybe.  However, try to read the ID3 tag off of one of these files.  Turns out they've stripped them of all information.  Google Music has stored all of this info in a private database inside the app.  That sucks.

So, yes.  You can use iSyncr to sync these mp3 files into iTunes, but it wouldn't be terribly useful.  You would have to manually add all of the song information after listening to it.  Hopefully Google opens up the Music API, or at least doesn't take steps to make sure you have to use their player to play pinned files.  A moderately decent intern could reverse these changes to the music app in a day.