Randomly getting signed out?

Hmm. You already downloaded your profile, so you can try the above, but I don't think it will work. What I suggest doing is deleting the Title Update for Black Ops 2 by going to Settings>System>Storage>[drive game is saved on]>Black Ops 2. You should see
the Title Update in there. Delete only that. Then go back to the main Storage bit and clear your system cache by following the instructions above. Launch BLOPS2 again, get the title update, and see if it runs fine after that.

It looks like your account is connected to a certain controller this can be done

in the Xbox Settings and I believe the Xbox Accessories app. Try booting up the console here the account is currently linked and un-link it to get rid of the error. That what I had to do to fix this issue hope this helps!

Sounds like the local profile is out of date or corrupted, and when an app or routine asks for data from it, it causes the involuntary disconnection. I'd start by clearing the system cache (it's just temporary files plus game updates, automatically reloaded)
by:

1) Please do Settings | System Settings | Storage

2) Move over any hard drive or memory unit (not the Cloud) & press (Y) on your controller & pick Clear System Cache.

3) When asked to confirm storage device maintenance, choose Yes.

4) Repeat that twice more

* Re-download *

5) Tap the silver Xbox Guide button

6) Select Download Profile (and if you don't see it, press the blue X button to make sure you're not signed in) and proceed with Profile Download.

7) Enter the Microsoft Account addresse for your Xbox LIVE gamertag when asked.

8) Enter the password for the Microsoft Account.

8A) If your account is an Xbox LIVE child account, the parent’s Windows Live ID & password is required as well to download your profile.

9) Pick the storage device for your profile (choose Hard Drive if you have one, or Memory Unit if you don't have a Hard Drive).

10) When the download completes you will be asked if you want the console to remember your password. If it's your home console, I'd say yes, otherwise I'd think carefully about leaving my profile unprotected on somebody else's Xbox!

11) Sign in, and retry.

I'm not sure about a cure, but this is a cross platform platform problem with the game itself, not your profile.

It can also, sometimes, be a profile issue. When an app or routine asks for data from it, it causes the involuntary disconnection. I'd start by clearing the system cache (it's just temporary files plus game updates, automatically reloaded) by:

1) Please do Settings | System Settings | Storage

2) Highlight any hard drive or memory unit (not the Cloud) & press (Y) on your controller & pick Clear System Cache.

3) When asked to confirm storage device maintenance, choose Yes.

4) Repeat that twice more

* Re-download *

5) Tap the silver Xbox Guide button

6) Select Download Profile (and if you don't see it, press the blue X button to make sure you're not signed in) and proceed with Profile Download.

7) Enter the Microsoft Account addresse for your Xbox LIVE gamertag when asked.

8) Enter the password for the Microsoft Account.

8A) If your account is an Xbox LIVE child account, the parent’s Windows Live ID & password is required as well to download your profile.

9) Pick the storage device for your profile (choose Hard Drive if you have one, or Memory Unit if you don't have a Hard Drive).

10) When the download completes you will be asked if you want the console to remember your password. If it's your home console, I'd say yes, otherwise I'd think carefully about leaving my profile unprotected on somebody else's Xbox!

11) Sign in, and retry.