After trying your solution I continued to receive the following error message:
"Error writing to file C:/Program Files(x86)/QuickTime/QTTask.exe. Verify that you have access to that directory"
With the typical Abort, Retry or Ignore options. I tried all of the options, Retry simply repeated, Abort cancelled the installation, and Ignore seemed to move the installation forward until I got the following error message: "Error writing to file C:/Program Files(x86)/QuickTime/PropertyPanels\PanelHelperBase.qpa. Verify that you have access to that directory" with no option available other than Cancel which removed all the files installed up to that point.
Frustrated with the solutions offered after an exhaustive google search "unistall all Apple applications (except iTunes) and then re-run the most current installer" or the ever so helpful "system wipe and restore" solution. I finally came across a solution from 2010 that worked for me.
I was able to install QT 22.214.171.124 for windows 7 by changing the default installation folder "quicktime" to a new folder that I named "aquicktime". Then changing the folder name back to QuickTime, and running the installer one more time choosing the Repair option and QT 126.96.36.199 has worked wonderfully ever since.