Install OS X recovery mode. Can't log in

Solution 1:

The version of the operating system of the recovery partition is probably lower than 10.11.0.

According to Apple you have to append the auth code to the password then:

What if I use two-factor authentication on a device running older software? If you use two-factor authentication with devices running older OS versions, you might be asked to add your six-digit verification code to the end of your password when signing in. Get your verification code from a trusted device running iOS 9 and later or OS X El Capitan and later, or have it sent to your trusted phone number. Then type your password followed by the six-digit verification code directly into the password field.