MacOS – Finally upgraded the Mac, now I can’t install security update

I finally upgraded from Snow Leopard to Mountain Lion (I didn't trust my older iMac (iMac11,3) to run the newest OS X) but now I can't install an important security update, the Security Update 2015-006 1.0.

I'm totally at a loss. I even booted up into the recovery mode, repaired all disk/permissions (didn't see anything serious anyway) and then re-installed Mountain Lion all over again. I'm still getting an error when I try to install it though.

It downloads just fine, and gets to the end of the install process after rebooting, but just gives an "unexpected error."

How can I fix this?

EDIT:
I tried downloading and installing the standalone update, but it didn't work either. Here are (what I understand to be) the output logs from the end of the installation process, when it fails:

Oct  6 05:16:42 homebase-iMac.local com.apple.kextd[12]: kextcache error while updating / (error count: 28)
Oct  6 05:16:43 homebase-iMac.local mds[761]: (Normal) ImportServer: Software update complete
Oct  6 05:16:43 homebase-iMac.local com.apple.SecurityServer[15]: Succeeded authorizing right 'com.apple.ServiceManagement.daemons.modify' by client '/usr/libexec/UserEventAgent' [11] for authorization created by '/usr/libexec/UserEventAgent' [11] (100012,0)
Oct  6 05:16:44 homebase-iMac.local Installer[928]: CoreAnimation: warning, deleted thread with uncommitted CATransaction; set CA_DEBUG_TRANSACTIONS=1 in environment to log backtraces.
Oct  6 05:16:44 homebase-iMac.local Installer[928]: The Installer encountered an error that caused the installation to fail. Contact the software manufacturer for assistance.
Oct  6 05:16:48 homebase-iMac.local com.apple.kextd[12]: /: kextcache has had enough tries; not launching any more