How to upgrade Avatar ROM's old 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.