An 800 x 800 image contains 640,000 pixels.
A 2048 x 2048 image contains 4,194,304 pixels, so about 85% of the pixels in the enlarged image have to be invented.
So this is never going to look good.
I managed to get a reasonably good result with Topaz Gigapixel AI and Topaz Sharpen AI. (you can get a 30 day free trial for both of them) I first enlarged the image to 2048 x 2048 in Gigapixel AI, using the Compressed option, with Suppress noise and Reduce blur set to zero.
Then I sharpened the enlarged image with Sharpen AI, using the Focus option, with Sharpness set to 70 and Noise suppression set to 50.
I just processed the original jpg, and had to save it twice, which is not good for image quality.
The jpg format uses destructive compression every time you save, and if you keep saving a jpg over and over, it will eventually start to disintegrate.
So whichever method you use, I suggest that you save all jpgs as Tiff, and edit the Tiffs.
And here's the original: (800 x 800)
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