Step 1: Unlock bootloader of and root your Galaxy Nexus. Additionally, flash ClockworkMod Recovery.
Step 2: Go to Settings > About on your Galaxy Nexus and note the Android version and build number.
Now, with that information in hand and knowing whether your GSM Galaxy Nexus is a yakju or takju variant*, go the following thread over on [XDA-Developers] forum and download the appropriate .zip file to update to Android 4.1.1 Jelly Bean (Build: JRO03C).
Step 3: Transfer the .zip file to the root of the /sdcard directory on your Galaxy Nexus.
Step 4: Now reboot into ClockworkMod (CWM) Recovery.
Step 5: Take a nandroid backup.
Step 6: Choose install zip from sdcard > choose zip from sdcard > [file you downloaded in Step 2]. This will flash the Jelly Bean update on your Galaxy Nexus.
Step 7: Once flashing process is complete, select reboot system now from CWM’s main menu.
