Redirect top level domain alias to specific og group node

I have my main site (alpha.com) that has a number of organic groups. I'd like to create a top level domain name (beta.com) that gets directed to a specific organic group group content type page (alpha.com/node/10). It would be really great if the URL to showed "beta.com" for any page/content within the "beta group" but switched to "alpha.com" for anything outside of the beta group content. Is this possible?

I tried redirecting through htaccess file but (1) I get thrown into a loop (beta.com/node/10/node/10...) and (2) I found out that this won't work with my host because they use nginx.