It Does count towards them.
That’s how I got this week’s bounty done
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I've never had it progress my bounties in strikes or anything.
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so did mine i ran a horror story with osmosis, solar sniper and solar grenade launcher dawnblade in a solo nightfall and most of those kills came from the horror i ran out of heavy and special 30 kills into the 150. so i have that same bounty for my alt i'll switch and run horror story to check for sure.