Can you cast a sorcery-speed spell during an opponent's turn if you have been given an extra turn effect?

Sorcery speed is defined by CR 307.1: a sorcery (and any spell or ability that can only be cast "as a sorcery") may be cast only during your own main phase, while the stack is empty. An extra turn simply gives you an additional turn structure — it does not change or suspend the timing rules.

When an effect like Time Walk or Time Warp grants you an extra turn, that extra turn occurs after the current turn ends. During that extra turn, you are the active player, so all normal timing permissions apply to you — including the ability to cast sorceries during your main phase. The critical point is that the extra turn is yours; it is not inserted into an opponent's turn.

You cannot cast a sorcery-speed spell during an opponent's turn under any normal circumstance. No extra turn effect changes this, because extra turns don't happen inside an opponent's turn — they happen after it. CR 500.1 establishes that each turn follows the same phase structure regardless of who generated it.

Concrete example: It is your opponent's combat phase. You have a Time Warp in hand. Even if you cast it right now (at instant speed, which you legally can), you would not be able to cast sorceries on your opponent's turn. Your extra turn begins after your opponent's turn ends, and you can cast sorceries freely during the main phases of that extra turn.

The only way to cast a sorcery-speed spell during an opponent's turn is through an effect that explicitly grants that permission, such as cards reading "you may cast sorceries as though they had flash." Extra turn effects do not grant any such permission (CR 307.1, CR 116.1a).