I have a file on my desktop that, when double clicked, takes me to a website. I created this by dragging the website's favicon to my desktop. The resulting file looks like this:

Is it possible to attach a fancy icon to this web link file so that, when I email the file to other people on various platforms, the link looks like the fancy icon instead of the generic image above? I'm able to change the icon on my desktop without a problem. The issue is that the new icon doesn't "stick" when I try to email it.
I currently have the icon in .icon format but could also create one in .icns or possibly anything else I need to do. Is this possible?

Is it possible to attach a fancy icon to this web link file so that, when I email the file to other people on various platforms, the link looks like the fancy icon instead of the generic image above? I'm able to change the icon on my desktop without a problem. The issue is that the new icon doesn't "stick" when I try to email it.
I currently have the icon in .icon format but could also create one in .icns or possibly anything else I need to do. Is this possible?