Hi Guys,
Your reply helped me as well. Thank you. I'm still struggling with the issue of rewriting /admin when using subpath.
When I click login icon I'm redirected with 302 to $host/admin not $host/fa/admin
This happened after I updated my FA.

location /fa/ {
        proxy_http_version 1.1;
        proxy_set_header X-Forwarded-Proto https;
        proxy_set_header X-Forwarded-Host $host;
        proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
        proxy_set_header X-Forwarded-Server $host;
        proxy_set_header X-Real-IP $remote_addr;
        proxy_set_header Accept-Encoding "";
        sub_filter       'action="/'  'action="/fa/';
        sub_filter       'href="/'  'href="/fa/';
        sub_filter       'href="/admin/'  'href="/fa/admin/';
        sub_filter       'src="/images'  'src="/fa/images';
        sub_filter       'src="/js'  'src="/fa/js';
        sub_filter_once  off;
        proxy_pass http://localhost:9011/;
    }
location ~^/(?<fusionPath>(oauth2|admin|ajax|login|password|js/identityProvider))/ {
            proxy_pass       http://127.0.0.1:9011/$fusionPath/;
            # https://fusionauth.io/docs/v1/tech/admin-guide/proxy-setup#how-to-use-a-proxy
            proxy_set_header X-Forwarded-Proto https;
            proxy_set_header X-Forwarded-Host $host;
            proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
            proxy_set_header X-Forwarded-Server $host;
            proxy_set_header X-Real-IP $remote_addr;
            proxy_set_header Accept-Encoding "";
            sub_filter       'action="/'  'action="/fa/';
            sub_filter       'href="/'  'href="/fa/';
            sub_filter       'src="/images'  'src="/fa/images';
            sub_filter       'src="/admin'  'src="/fa/admin';
            sub_filter       'src="/js'  'src="/fa/js';
            sub_filter_once  off;
        }