Other appsare terminated when I try to copy an auth code from the Google Authenticator I have the Google Authenticator app and I want to log into Stack Overflow with my Google account.

But if I leave the login page from app to look at the authenticator code, the SO application refreshes.

It can't use an application specific password. Any ideas?
You can't use an app specific passphrase. Stackexchange uses Google as an OpenID provider and that works with access tokens.

To continue/finish the OpenID process you need to be logged in to Google of course (using the regular 2-step passphrase+token method). You will then be asked if you want to grant access by Google and the app will get an access token.