How do you design a favicon and an avatar that match?
The small icon and the avatar are the simplified mark, the one recognised without reading the whole name. Do not shrink the full sign down to a handful of pixels.
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Look at the mark in a circle the size of a profile photo, on the phone. The avatar is the whole poster and in the chat list it is only a smudge. Two exports: the website icon and the profile circle, from the same short mark.