How to sync music bought on Google Play back into iTunes I have the Google Music manager set to watch my iTunes folder, so if I buy music on iTunes, it'll end up in my Google Play Music.

What about the other direction?

When I buy a song on Google Play, how do I set things up so that it will be added to my iTunes collection on my Mac?
I've bought one album on Google Play Music and because I've set in Preferences that I want monitor my iTunes folder, everything new in iTunes is uploading to GPM.
When I bought this album it started downloading to my iTunes library automatically but when I deleted in on iTunes (I needed to free some disk space) it hadn't appeared again. So I have to download manually whole album using Google Play extension in Chrome and add it to iTunes.
So I assume there is no "sync" or "push" to iTunes in GPM.

Recent introduction of Apple Music brings more confusion. if you are Apple Music abonent then:
1. When you buy some album on GPM and it will go to iTunes.
2. If you turned on iCloud Music Library the album start to upload to iCloud and songs start to being matched by Apple.
3. As long as you are subscribed to AMusic you may hear this album "from the cloud" on all your PC and iOS devices with exception of Apple TV for now and Android phones (will be possible in fall 2015).