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.
** It may be important to note that I had already uninstalled the previous version of QuickTime that I was attempting to upgrade before I came across the solution listed above. I wish I had found the solution earlier, because uninstalling the previous version of QuickTime wasn't necessary.