The Mac is on version 10.11.4 of OS X. I'm using Safari. I can download new version 188.8.131.52, double-click, get prompted for password and install starts, but progress bar gets stuck at 90%, then after a while quits with failure. I've already tried downloading the offline installer from the Mac troubleshooting guide and the issue persists.
I'll need to get some log files from you to troubleshoot further. Please read through the FAQ Where do I find the Flash Player installation log on the Macintosh? before doing anything. After reading through the FAQ do the following: