Can't update to a previous kernel

I tried to use a patched kernel version 6.0.12 to fix undetected sound module. I did make -j$(nproc) && sudo make module_install && sudo make install. These are the files inside the /boot after several sudo make install:

$ ls -l /boot
total 964108
-rw-r--r-- 1 root root    270644 May  1 20:49 config-6.0.12+
-rw-r--r-- 1 root root    271106 May  1 19:37 config-6.0.12+.old
-rw-r--r-- 1 root root    275729 Apr 12 22:02 config-6.2.6-76060206-generic
drwx------ 5 root root      4096 Jan  1  1970 efi
lrwxrwxrwx 1 root root        18 May  1 19:38 initrd.img -> initrd.img-6.0.12+
-rw-r--r-- 1 root root 796981300 May  1 20:49 initrd.img-6.0.12+
-rw-r--r-- 1 root root 133714610 May  1 18:35 initrd.img-6.2.6-76060206-generic
-rw-r--r-- 1 root root   6294956 May  1 20:49 System.map-6.0.12+
-rw-r--r-- 1 root root   6300193 May  1 19:37 System.map-6.0.12+.old
-rw------- 1 root root   7975326 Apr 12 22:02 System.map-6.2.6-76060206-generic
lrwxrwxrwx 1 root root        15 May  1 20:49 vmlinuz -> vmlinuz-6.0.12+
-rw-r--r-- 1 root root  10633952 May  1 20:49 vmlinuz-6.0.12+
-rw-r--r-- 1 root root  10638464 May  1 19:37 vmlinuz-6.0.12+.old
-rw------- 1 root root  13850208 Apr 12 22:02 vmlinuz-6.2.6-76060206-generic
lrwxrwxrwx 1 root root        19 May  1 20:49 vmlinuz.old -> vmlinuz-6.0.12+.old

I did run

sudo kernelstub -v -f -k /boot/vmlinuz-6.0.12+ -i /boot/initrd.img-6.0.12+

However, afterwards I saw that the kernel path from sudo kernelstub -p doesn't change:

sudo kernelstub -p
kernelstub.Config    : INFO     Looking for configuration...
kernelstub           : INFO     System information: 

    OS:..................Pop!_OS 22.04
    Root partition:....../dev/nvme0n1p3
    Root FS UUID:........56eca1a3-9638-4cd7-ac4b-f8d9aa48f455
    ESP Path:............/boot/efi
    ESP Partition:......./dev/nvme0n1p1
    ESP Partition #:.....1
    NVRAM entry #:.......-1
    Boot Variable #:.....0000
    Kernel Boot Options:.quiet loglevel=0 systemd.show_status=false splash
    Kernel Image Path:.../boot/vmlinuz-6.2.6-76060206-generic
    Initrd Image Path:.../boot/initrd.img-6.2.6-76060206-generic
    Force-overwrite:.....False

kernelstub           : INFO     Configuration details: 

   ESP Location:................../boot/efi
   Management Mode:...............True
   Install Loader configuration:..True
   Configuration version:.........3

After a restart, I saw that the uname -a is updated to 6.0.12+, but not the kernel and initrd image paths from kernelstub -p, which still shows as above paths..

$ uname -a
Linux pop-os 6.0.12+ #2 SMP PREEMPT_DYNAMIC Mon May 1 20:44:44 CEST 2023 x86_64 x86_64 x86_64 GNU/Linux

At the end, I can't see any expected improvement after the reboot. Since there is no grub-update in PopOS, I'm so confused and unsure how to switch the Linux kernel. There is no official documentation about this. The documentation https://support.system76.com/articles/kernelstub/ is still confusing since the kernel image path doesn't change after executing the command.

pop-os / linux

Can't update to a previous kernel #252