Not to be showing off here, but I have IRST 188.8.131.522 installed on my ASRock Z77 Extreme4 board:
I assume that the way I installed IRST 184.108.40.2062 is why I was able to succeed. More about that in a moment.
First, if we check the ReadMe on Intel's download page for IRST 220.127.116.112, it includes the Intel 7 series chipsets in the support list for this version of IRST:https://downloadmirror.intel.com/25165/eng/ReadMe.txt
Unless that document was not updated correctly, Z77 boards should be able to use IRST 18.104.22.1682. If that really is important is another question.
I installed IRST 22.214.171.1242 during a new Windows 10 installation, using the f6flpy-x64.zip type driver, using the Load Driver option in the Custom Installation option. (Why does Intel still use the "flpy", floppy disk name for this driver, that drives me crazy!)
The SATA Mode was set to RAID, and I did a "UEFI booting" Windows 10 installation (CSM set to Disabled in the UEFI/BIOS), on an Intel 750 AIC SSD.
I also had a new UEFI/BIOS version for this board installed, P2.90M, that includes NVMe support, in its Beta download section:http://www.asrock.com/mb/Intel/Z77%20Extreme4/?cat=Beta
Why am I using an Intel 750 in this board? (Why not?) Originally as a test to confirm it was possible to use an NVMe SSD as the OS drive in this board, with the new UEFI version. I'm a mod in ASRock's new forum, so I get to do things like this.
So why are others having problems installing IRST 126.96.36.1992 on Z77 boards? I'm not sure.
I can't get the IRST 188.8.131.522 Windows UI to install on my Z170 board, with a Samsung 950 Pro as the OS drive, with one SATA III SSD in that PC. The IRST 184.108.40.2062 installer just hangs during the IRST UI installation process. The 220.127.116.112 driver installs apparently, same thing happens with the IRST 15 versions from Station Drivers. That is another story...
I checked the IRST Option ROM version in the IRST UI System Report, which shows 18.104.22.1682. That should be the legacy (non-EFI) Option ROM, which was not updated in the UEFI/BIOS version I'm using. I don't know what version the EFI Option ROM is, should check that with UBU. I should be using the EFI Option ROM, which is necessary for a PCIe NVMe SSD, but that has never been a requirement for installing IRST in general.
I have the INF/Chipset files version 10.1.1.8 installed.
You could try installing the basic IRST 22.214.171.1242 driver in the f6flpy-x64.zip package, by doing a manual driver update in Device Manager.
There really should be nothing related to the Z77 chipset in IRST 14.x drivers. The big addition in IRST 14 is RAID support for NVMe SSDs, but I doubt that feature will work with the Z77 chipset. I believe NVMe RAID support only works with the Intel 100 series chipsets, like the Z170.
If you are doing a new Windows installation on your Z77 board, the trick might be installing the 126.96.36.1992 f6flpy-x64 driver during the installation. Would that work in AHCI mode? Don't know, never tried it.
Otherwise, I doubt you are missing anything by not having IRST 188.8.131.522 installed on your Intel 7 series chipset board.