Does CyanogenMod include a kernel? I've installed CyanogenMod 10.1 stable for S III LTE (i9305), and noticed that my kernel version is 3.0.64 (I expected it to be 3.4 or something higher).

As I've said in this question, the kernel I had before was also from the 3.0.* line.

Which is quite weird - why is my kernel always staying on that version?

Could it be that CyanogenMod doesn't come with a kernel, but that the kernel is flashed somewhere else?

If not, what's the reason for such a low version number? Is it because my phone isn't all that well-supported? Should I manually flash a new kernel for my phone?
Looking at the repository for that device, I can see that the specified kernel is built from source using the source found here and the cyanogenmod_i9305_defconfig configuration file.

You can tell that the kernel is not the stock one by going to the terminal app (or from adb shell) and typing uname -a, which will tell you the build date of the kernel.  If it's the same as the creation date for the whole OS, then you know it's a custom-built kernel just for CM.

The CM11 branch also uses the cm11 branch of this kernel repo...