I'm using Tasker on Android Lollipop (5.1.1). I've come to know that applications now can write to external SD card in Android 5.1.1 update but with the use of an API, for which the user needs to grant permission to the application. I have successfully done this with ES File explorer.
But, how do I give Tasker the permission to write to external SD Card?
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pm grant …
– but as you already pointed out, write-access to SD card with LP+ is covered by specific API calls, so an app must use those. If it doesn't, the dev must adjust. Not much a user can do about that (at least not without root).