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You could just send out a link to the .apk, it doesn't even need to be through the store.

Last I heard, Google supported completely discretionary developer-initiated refunds. IIRC, at least a couple of the high-profile apps that left the Amazon Appstore took advantage of that to ease the transition for their paying (i.e. not FAOTD) customers. With that capability, implementing a reasonable functional equivalent of a free promo code seems straightforward enough.

How about e-mail?

I've always wondered, what would happen to Google's 30% cut if they implemented promo codes, and I used it to send a few copies of my paid app?

That's why Apple limits the number of promo codes you can issue to 50 per version last time I checked, so you can't sell the codes to users and avoid giving up the distribution cut.