As the title says play music hasn't been updated to support storage formatted as internal when you download and save music, reason I posted here is because I formatted my storage on the Pure. Bummer.
The Google music app doesn't give the option to move to SD card, at least not on Lollipop. Amazon music does however, if you use it. I do use it and have put it on my SD card. It also gives the option to put any new downloaded music straight to the card, so you're not using your phone's internal storage at all.
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The app itself? I have the Google music app configured to put music on my SD card on lollipop. Never tried to move the app.