What is the exact error you are getting? Is it 'insufficient permissions' or 'failed to register'?
'insufficient permissions' is usually returned when the user account is not an Administrator. Flash Player requires Admin permissions to install, therefore the user must either be an Adminstrator, or provide Administrator credentials to install.
'Failed to register' error means there are permissions issues with either registry items or Flash Player files and it is preventing the installation.
Please provide the FlashInstall.log file saved at C:\Windows\System32\Macromed\Flash using the instructions The specified item was not found.. If this is a 64-bit OS, the FlashIntall.log file saved at C:\Windows\SysWOW64\Macromed\Flash is also required.