3 huge peaks in the heart of the wild Georgian mountains!
At over 3600 metres, the Bethlemi Hut is deep in the Caucasus mountains, just to the north of the legendary Gudauri ski area. Gudauri itself is home to some epic mountain terrain (as evidenced here https://fatmap.com/guidebooks/32914/the-best-of-georgia-:-email@example.com,44.4115510,26855.6330866,-89.9999999,0,2715.7940490,normal) but things get even wilder as you head north and make the long trip into this hut.
Once there we've described the routes to 3 summits, 2 of which are ascended on skis and 1 of which is an adventurous alpine climb. By heading into the Bethlemi Hut in April or May, and with a bit of luck with weather and conditions, it should be possible to do all 3 trips over the course of a few days.
Of the 2 ski trips, there's no doubt that "Mkinvartsveri/Kazbek (5054 m) SE Face" is the tougher tick - it's higher, steeper and would be considered a serious challenge even in a much more developed ski destination. The other ski line - "Maili Khokh (4600 m), via Spartak (4517 m)" is technically easier but is still incredibly remote and will require you possess a full mountain skillset in order to move in complete autonomy through a truly wild place.
The final route described here - "Ortveri (4365 m) NE Face" - is a brilliant little adventure, and is a good acclimatisation route for the 2 tours listed above. Elite skiers may choose to ski down the route, but conditions would need to be perfect, as would your skiing technique...!
If you're aiming to do all of these routes in one go then head into the Bethlemi Hut with a week's worth of food and enjoy 3 huge mountain adventures, and some well-earned rest days savouring the solitude.