Buildroot kernel module reports invalid module format

I am working on building a test kernel module in Buildroot for a Raspberry Pi target. My host machine is x86, running Ubuntu 22.04.1. I have been following along with several tutorials, including this one and this one and this one.

My buildroot system boots, and I can see my compiled kernel module at /lib/modules/5.15.92-v7l/extra/kernelmoduletest.ko.xz.

If I attempt to run insmod from the home directory, my module cannot be found. But, if I navigate to where it is installed and run insmod, I get 'invalid module format'.

# pwd
/root
# insmod kernelmoduletest.ko.xz 
insmod: can't insert 'kernelmoduletest.ko.xz': No such file or directory
# cd /lib/modules/5.15.92-v7l/extra/
# ls
dtbocfg.ko.xz           kernelmoduletest.ko.xz
# insmod kernelmoduletest.ko.xz 
insmod: can't insert 'kernelmoduletest.ko.xz': invalid module format

Looking into the invalid module format error seems to suggest that the kernel module was compiled for the wrong kernel?

I ran dmesg to get more info, and I found this:

[    3.820114] udevd[136]: starting version 3.2.11
[    3.852896] udevd[137]: starting eudev-3.2.11
[    4.497392] bcmgenet fd580000.ethernet: configuring instance for external RGMII (RX delay)
[    4.507484] bcmgenet fd580000.ethernet eth0: Link is Down
[    8.635423] bcmgenet fd580000.ethernet eth0: Link is Up - 1Gbps/Full - flow control rx/tx
[   33.754917] cam-dummy-reg: disabling
[   66.809984] Invalid ELF header magic: != ELF
[   66.814488] Invalid ELF header magic: != ELF

How do I compile my module correctly so it will run with insmod?

Here is my setup information for my kernel module, called kernelmoduletest:

buildroot/package/kernelmoduletest/Config.in:

config BR2_PACKAGE_KERNELMODULETEST
    bool "kernelmoduletest"
    help
        kernelmoduletest package.

        Kernel module test. Just a hello world package that runs as a kernel module.

buildroot/package/kerneltestmodule/kernelmoduletest.mk:

################################################################################
#
# kernelmoduletest package
#
################################################################################

KERNELMODULETEST_VERSION = 1.0
KERNELMODULETEST_SITE = package/kernelmoduletest/src
KERNELMODULETEST_SITE_METHOD = local# Other methods like git,wget,scp,file etc. are also available.

#KERNELMODULETEST_MODULE_SUBDIRS = driver/base
KERNELMODULETEST_MAKE_OPTS = KVERSION=$(LINUX_VERSION_PROBED)



define KERNEL_MODULE_BUILD_CMDS
        $(MAKE) -C '$(@D)' LINUX_DIR='$(LINUX_DIR)' CC='$(TARGET_CC)' LD='$(TARGET_LD)' modules
endef


#ifeq ($(BR2_PACKAGE_LIBBAR),y)
#KERNELMODULETEST_DEPENDENCIES += libbar
#KERNELMODULETEST_CONF_OPTS += --enable_bar
#KERNELMODULETEST_MODULE_SUBDIRS += driver/bar
#else
#KERNELMODULETEST_CONF_OPTS += --disable_bar
#endif

$(eval $(kernel-module))
$(eval $(generic-package))

buildroot/package/kerneltestmodule/src/kernelmoduletest.c:

#include <linux/module.h>
#include <linux/kernel.h>

MODULE_LICENSE("GPL");

static int myinit(void)
{
    printk(KERN_INFO "kernelmoduletest hello init\n");
    return 0;
}

static void myexit(void)
{
    printk(KERN_INFO "kernelmoduletest hello exit\n");
}

module_init(myinit)
module_exit(myexit)

buildroot/package/kerneltestmodule/src/Makefile:

obj-m += kernelmoduletest.o
kernelmoduletest.o += -DDEBUG

all:
    #make ARCH=arm CROSS_COMPILE=arm-linux-gnueabihf- -C $(KDIR) M=$(PWD) modules
    $(MAKE) -C '/lib/modules/$(shell uname -r)/build' M='$(PWD)' modules

clean:
    #make ARCH=arm CROSS_COMPILE=arm-linux-gnueabihf- -C $(KDIR) M=$(PWD) clean
    $(MAKE) -C '$(LINUX_DIR)' M='$(PWD)' clean