Split-Screen Xbox Live in Call of Duty: Modern Warfare II (2022)?

No. Call of Duty: Modern Warfare II (2022) and MWIII (2023) do not support Split-Screen Online Multiplayer on Xbox (or PlayStation).

Mode	Split-Screen?
Campaign	❌ No (Single Player Only)
Spec Ops / Raids	❌ No (Online Co-op Only)
Multiplayer (Online)	❌ No
Private Match (Local)	✅ Yes (2 Player) — Offline Only (Bots / 1v1)
Zombies (MWIII)	❌ No

CoD 4 / WaW / MW2 (2009) / BO1 / MW3 (2011) / BO2 / Ghosts: Supported 2-Player Split-Screen Online (Xbox 360 / PS3).
AW / BO3 / IW / WWII / BO4 / MW (2019): Removed Split-Screen Online (Engine/Performance).
MWII (2022) / MWIII (2023) / BO6: No Split-Screen Online. Local Private Match Only.

Workaround: Two Consoles + Two Accounts + Two Copies (or Game Share) + LAN / Online Party.

Xbox Live