By "host" do you mean name registrar or is there also web content or something else you're hosting?

The limitation described is one that Google has implemented, not one that MS Outlook has.

Other email clients deal with this correctly, though.

I assume you mean a DNS registrar, not a host. I've used joker.com for about ten years now. They're based in Switzerland (or is it Germany?). About $10/year for .com/net/org domains.

Certainly differently. "Correctly" is debatable.

It is reasonable to also see it as a limitation of other clients that they do not correctly display the "on behalf of" details where present.
Odds are he won't need to move the registration, just update a few DNS records.

I guess it depends on what you mean by "correctly" - it treats the @mydomain.com domain sender as a delegate for the @gmail.com sender, and shows that. It's a bit goofy because that feature is really intended for when a secretary/assistant/whatever sends on your behalf, but I'm not sure it's inherently incorrect.

Annoying? Abso-f'in-lutely.

I had the same "sent on behalf of" issue with my cellular modem back when Sprint still had SMTP. The most annoying thing was if that an email I sent to someone was forwarded to someone else, and that third person replied to it, it went to my <[email protected]>.

Does Google Apps have a workaround for that? I've been poking through their FAQ, but nothing is jumping out at me.

If you are hosting the domain on google apps then it will just send from you. If you have access to a secured (TLS/SSL) SMTP server from the real host you can configure google to send using that SMTP and it will skip the on behalf of.