I didn't think there was such a thing as a "league 2 referee". They are EFL referees and officiate throughout the three leagues, although granted the 'better' ones do officiate more championship games and the not so good ones get the league 1 & 2 games. I'm sure Deepcut will confirm if this is correct.
Premier League - Select Group
Championship - Select Group 2
Leagues One and Two - Level 1
National League - Level 2a
But these can be appointed one step up, especially if they are potential promotion candidates.