Where have you activated hotlink protection?
Typically, hotlinking looks at the referrer that a web browser sends as part of its request and ensures that it matches some rule before returning the file. If a person is looking at an e-mail that references your web site's address for the avatar picture, it's going to hit your web server as a request from mail.google.com or similar, which will probably kick in the hotlinking protection through whatever mechanism you've enabled it.
I don't recall a hotlink setting in IPS, so I'm guessing this is a web server configuration or perhaps CDN?
If you want to allow certain images to be served and bypass the hotlinking rules, you'd have to work with whatever is providing that service to find out how to define exceptions. With Cloudflare as an example, you can place images in special paths that indicate they're okay to be hotlinked, while other images that you don't want hotlinked aren't okay. You could achieve this with IPS by defining a file storage location for the image types you'd like to allow to be hotlinked, that matches the special path rules they use.
If you want to embed the images in the e-mail, I don't think that's an out of the box feature, but perhaps some custom development could achieve that.
Alternatively, you could set some exceptions perhaps (again, depending on the service) to allow some popular mail providers to get past the hotlinking rule, yet that might be a challenge and would not address the issue with thick client e-mail programs that may still get a broken image (again, depending on how you're blocking them).