I tried for hours to install Flash Player 16 on Mac OS X 10.5 using the suggestions I found. The automatic update function was useless. I never got past the retrieval stage; nothing downloaded for me. I tried the uninstall and reinstall of Flash Player with NO LUCK. Here is how I eventually was able to get Flash Player 16 to install:
1] I DID delete the old flash player - NO idea if this is necessary, since my next step was replacing the mach_kernel file.
2] I downloaded Pacifist ($20 shareware - but might work on trial basis). I used Pacifist to search and identify the mach_kernel file from an original Mountain Lion installation disk (should also work fine for an install image). I then instructed Pacifist to install the original mach kernel file on my system through the replace function. I rebooted the system so the New "old" mach_kernel file was now the operational Unix foundation for the computer.
3] Still no luck using the downloaded dmg install files from Adobe. I then went to the troubleshooting Adobe Systems file and chose the last option which I believe was install directly: Flash Player for Firefox and Safari- NPAPI. This WORKED and I duplicated it on another machiine. Here is the link:
Suggestion: As a first step, delete the old flash player, reboot and then try STEP 3 and avoid the downloaded DMG Adobe files. I suspect this may work but if it doesn't then go through these 3 steps...at least for Mountain Lion. I'd be interested in knowing whether Mavericks users have success with this.
Step 1 alone Deleting and trying to reinstall flash player DID NOT WORK FOR ME. Nothing retrieved past the dmg installer. Also reboot when you are done with the steps.