ETA: This was meant to be a reply to a PP about how pointless him descending to the hut was, in terms of getting help.
What help could he get there? There were unlikely to be people. There would've been no help. He could have got help by calling on his phone while staying with her, which did still have battery at the point they urgently needed help. They had been stopped climbing for hours by 12.30 am, when he called the emergency services to say they were fine – all he had to do was say they needed help then, which they clearly did.
And if he was well enough to hike over the summit and over an hour down the mountain, then when she was unable to go on he would've been well enough to get them both into a sheltered position, wrap them both in the emergency blankets with her in the bivy sack, and call for help.
From what I've read about how she was found, hanging by a rope feet down off a rock face with her crampons having loosened, I wouldn't be surprised if she was struggling and injured and being 'encouraged' to press on by him as the only option left to them, because he wanted to summit and not be rescued, and so, in her unsuitable footwear, injured and exhausted, she ended up falling.
If that happened, I can imagine in that weather, as tired as he was, he couldn't pull her back up, panicked, tried to tell the authorities everything was fine at half midnight, and then in the end just left her to die. He probably hoped they'd assume she fell while hypothermic and confused after he left her there to 'get help'. It would also explain why she wasn't in her emergency blanket or bivy sack.
That seems most likely to me, anyway.