If you have downloaded the game from Steam, search your Steam library for "Black Ops II", there should be 3 apps for it. Make sure you have the app of the mode you want to play installed.
If you have the mode installed and the problem still occurs, then restart Steam and repair the mode's game files: https://plutonium.pw/docs/repair/
If you have attempted to repair game files but it still does not work, check that the folder you selected in the Plutonium launcher is actually the folder which Steam is currently using for the game by comparing the paths.
For checking the selected folder in the Plutonium launcher, follow either of these:
Open the launcher, select the game, click "Game Settings", and check the "Game Path".
Press Win+R, paste this into the window that opens: %localappdata%\Plutonium, press Ok, open config.json, then check the path set for "t6Path".
For opening the folder Steam uses for the mode: right-click the mode from your Steam library > Manage > Browse local files.
Browse local files
If none of the previous suggestions have worked, follow this:
Go to your game folder. This can be done by finding the game on your Steam library, right-clicking it, then selecting Manage > Browse local files.
Exit out of Steam and make sure it is no longer running. Closing Steam's window is not enough to shut it down.
From the file path address for the game folder, click on steamapps.
Delete the game's appmanifest files. The following are the appmanifest files for BO2:
appmanifest_202970.acf
appmanifest_202990.acf
appmanifest_212910.acf
You should find at least one of those files in your steamapps folder. Do not worry if one or two of those files do not exist.
Reopen Steam. The game's mode should now appear as uninstalled when you check it on your game library and should now say "install" instead of "play".
If the mode does not appear as uninstalled after doing the previous steps, restart your PC.
Click the "install" button for the mode. This will verify the integrity of the installed files and will install any missing files that were not originally installed.