I used to do Ice Rescue (Trained & Certified through Dive Rescue International) when I was stationed on the Great Lakes. First and foremost, Telemark has it right. As hard as it will be, you must somehow notify EMS, even if it means leaving the scene. There’s no sense in the both of you dying, which is likely in this scenario.
As to the hypothetical “You have to try absent any help” scenario, the only things you can do have already been mentioned, just a few things to add.
First, dress as appropriately as you can. If you have extra clothes to throw on, do so.
Then, distribute your weight over a greater area. Crawling/sliding will do this absent any “sled” or raft device. Take something that floats if you can find it. If you have anything that could be used as line, grab it. Bring a hammer, hook, anything you can find.
Make some type of ice picks for each hand to aid in moving, and most importantly, to aid in getting out of the water should you fall through. Screwdrivers are excellent for this, but pointed sticks will do if that’s all you have.
Choose your route carefully. The most direct route may not be the best. If you have any knowledge of ice, think about that before you go. Clear ice is the strongest. Ice with snow on it should never be trusted. IIRC - 2-4 inches of good ice is plenty to support your weight. Take your time, and take no chances.
If you make it to the victim, now comes the hard part. Most likely, you will be laying on your stomach at the waters edge, with the victim submerged up to the chest holding on to the ice edge. Establish comms, and find out how bad they are. Talk positively and all that.
If you were able to bring a hook type of tool (hammer) or a line, try to get the victim to lasso themself or hook themself around the waise area. Try to get them to pull up out of water and assist with the waist area to “roll” them onto the ice. If they have the strength (unlikely) they can use the “ice picks” to try and pull themself out.
Go back slowly the way you came if conditions haven’t changed.
All in all, this is a very dangerous scenario to attempt this on your own without help.
One point to remember with cold water victims - they’re not dead until they’re warm and dead. What that means is this: Even if they drown, all is not lost. Cold water near drowning is a phenomenon that rescuers are well familiar with. When humans drown in cold water, there is a “golden hour” in which it’s possible to revive them. Last I heard, this golden hour may even be longer. This is due to the “mammalion dive reflex”. I won’t go into the specifics, but it works better in children than adults, women than men, drunks than sober, etc. People have been underwater for long periods of time, and revived without any brain damage or other permanent effects. The point being - get help first. Even if they drown, they can still be possibly saved. Don’t risk your own life as well.
I apologise for the spelling/typos. I’m at work, on a laptop.