eboot is the execution format that the psp can play at firmware 1.5 or custom.
Kernel is kinda like a firmware for your psp.
1-download all your updates from
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2-Put them in the game150 folder
3-Load it up and that's it. Just make sure it doesn't power off while it's updating or it could cause a brick. u no wun dat. Just charge it while it's upgrading and you'll be alright.
PS: I remember that alex talked to me about an auto update feature. Have you tried booting the psp in debug mode by holding L while booting it up. Disble psp automatic update then go where the normal update button normally is, click on it and it should connect you to darkalex's update hosting website, from there you can download the newest update.