How to upgrade Avatar ROM's old kernel? I've got an S3 LTE version (GT I9305), with CyanogenMod (Avatar) installed.

I've tried nightlies too, however, I'm still running Linux 3.0 (shown in about in settings). And I've seen CM screenshots where they're running 3.4.

Why is my phone stuck even when I update? How can I get a recent kernel?
I couldn't find any details on this XDA thread regarding Avatar ROM for the S3. However, the Sony Xperia Z thread thread mentions that it has a Linux kernel of 3.0.x. So, I assume it's the same for the S3.

While Avatar is based on CM10.1, it's possible that the developer used a different kernel (lower than 3.4). So, don't compare it with pure CM ROMs that are using the 3.4 kernel. I would suggest going through the first link I gave and ask for help there about upgrading your kernel.