There are very few rules associated with action surge, so this should be easy.
When you use extra attack, is it still your turn between your two attack rolls? Yes. Then by RAW you can use action surge at that point.
Without an explicit limiter elsewhere in the ruleset, restricting the timing of that additional action is clearly a ruling/house-rule, not RAW.
Can anyone cite a rule related to the restricted timing of the additional action that I'm missing?
Action Surge
Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On your turn, you can take one additional action.
When you use extra attack, is it still your turn between your two attack rolls? Yes. Then by RAW you can use action surge at that point.
Without an explicit limiter elsewhere in the ruleset, restricting the timing of that additional action is clearly a ruling/house-rule, not RAW.
Can anyone cite a rule related to the restricted timing of the additional action that I'm missing?