I downloaded the driver for the card from the Asix website and followed the installation instructions provided by Asix. I was able to compile the driver without any issues by executing "make". But after I executed "sudo make install" it failed:

$ sudo make install
rm -f *.mod.c *.o *.ko .*.cmd *.symvers
make -C /lib/modules/5.15.0-131-generic/build/  M=/home/nostromo/Downloads/work2 modules
make[1]: Entering directory '/usr/src/linux-headers-5.15.0-131-generic'
  CC [M]  /home/nostromo/Downloads/work2/ax99100x_spi.o
  CC [M]  /home/nostromo/Downloads/work2/ax99100x_sp.o
  LD [M]  /home/nostromo/Downloads/work2/ax99100x.o
  CC [M]  /home/nostromo/Downloads/work2/ax99100x_pp.o
  CC [M]  /home/nostromo/Downloads/work2/parport_pc.o
  MODPOST /home/nostromo/Downloads/work2/Module.symvers
  CC [M]  /home/nostromo/Downloads/work2/ax99100x.mod.o
  LD [M]  /home/nostromo/Downloads/work2/ax99100x.ko
  CC [M]  /home/nostromo/Downloads/work2/ax99100x_pp.mod.o
  LD [M]  /home/nostromo/Downloads/work2/ax99100x_pp.ko
  CC [M]  /home/nostromo/Downloads/work2/parport_pc.mod.o
  LD [M]  /home/nostromo/Downloads/work2/parport_pc.ko
make[1]: Leaving directory '/usr/src/linux-headers-5.15.0-131-generic'
gcc -pthread select_BR.c -o select_BR
gcc -pthread advanced_BR.c -o advanced_BR
gcc -pthread gpio_99100.c -o gpio_99100
gcc -pthread -Wall 9bit_test.c -o 9bit_test
gcc -I /usr/include spi_test.c -o spi_test
rm -f *.mod.c *.o .*.cmd *.symvers *.mod
rm -rf .tmp_version* *~
rm -rf Module.markers modules.* 
make -C /lib/modules/5.15.0-131-generic/build/  M=/home/nostromo/Downloads/work2 modules_install
make[1]: Entering directory '/usr/src/linux-headers-5.15.0-131-generic'
arch/x86/Makefile:142: CONFIG_X86_X32 enabled but no binutils support
cat: /home/nostromo/Downloads/work2/modules.order: No such file or directory
  DEPMOD  /lib/modules/5.15.0-131-generic
Warning: modules_install: missing 'System.map' file. Skipping depmod.
make[1]: Leaving directory '/usr/src/linux-headers-5.15.0-131-generic'
cp ax99100x.ko  /lib/modules/5.15.0-131-generic/kernel/drivers/tty/serial
su -c "/sbin/depmod -a"
modprobe ax99100x
modprobe: ERROR: could not insert 'ax99100x': Exec format error
make: *** [Makefile:77: install] Error 1

The warning message "CONFIG_X86_X32 enabled but no binutils support" said that binutils are missing but they are installed:

$ sudo apt-get install binutils
Reading package lists... Done
Building dependency tree... Done
Reading state information... Done
binutils is already the newest version (2.38-4ubuntu2.6).
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

$ depmod ax99100x
depmod: ERROR: Bad version passed ax99100x.ko

$ dmesg | grep modules
[31484.777507] module: x86/modules: Skipping invalid relocation target, existing value is nonzero for type 1, loc 00000000ff3c1226, val ffffffffc52e7b30

$ sudo lspci -v
03:00.0 Serial controller: Device 1c00:3050 (rev 10) (prog-if 05 [16850])
    Subsystem: Device 1c00:3050
    Flags: fast devsel, IRQ 18
    I/O ports at d000 [size=256]
    Memory at d2100000 (32-bit, prefetchable) [size=32K]
    I/O ports at d100 [size=4]
    Expansion ROM at df600000 [disabled] [size=32K]
    Capabilities: [60] Power Management version 3
    Capabilities: [68] MSI: Enable- Count=1/32 Maskable+ 64bit+
    Capabilities: [80] Express Legacy Endpoint, MSI 00
    Capabilities: [100] Advanced Error Reporting
    Kernel driver in use: parport_pc
    Kernel modules: parport_pc, parport_serial

I figured out that I have disable secure boot to be able to load the kernel module, so I did:

$ mokutil --sb-state
SecureBoot disabled

I did but I still could not make it to work.

Here the version of my linux installation (Linux Mint):

$ uname -a
Linux lv426 5.15.0-131-generic #141-Ubuntu SMP Fri Jan 10 21:18:28 UTC 2025 x86_64 x86_64 x86_64 GNU/Linux

Since more than one week I am struggling with this problem. Has anybody an idea how to fix it?