I keep trying to update, it finishes to download but it does not install. no error message.

The updaters on Apple's download site are not full installers, so attempting to install macOS 10.15.7 update on Mojave would expect some earlier version of Catalina already installed older than that last version, and as you witnessed, fail.

In order to upgrade from Mojave to Big Sur, you would need to have a Late-2014 model 27-in iMac or newer, and to Monterey, a Late-2015 model or newer. If your iMac is a Late-2013 model, it can only upgrade to Catalina.

Remove the downloaded updater(s).

Using exclusively the Safari browser, see How to get old versions of macOS - Apple Support to obtain the pertinent full installer download from the Mac App Store. It will install in your /Applications folder and automatically launch offering you the opportunity to perform the Upgrade or quit the installer. Never perform an upgrade without a last Time Machine backup of the current operating system and then set that drive aside, not to be used as a Time Machine drive by the new operating system. It is your get out of jail free card.

Any version of macOS after Mojave will not support 32-bit applications, so I suggest that before the upgrade, you run the free Go64 application that will list the 32-bit application dependencies. For those third-party applications, printer,s and scanner drivers, you will need 64-bit replacements from those vendors, provided 64-bit versions exist and are compatible with your upgrade choice.

macOS Big Sur - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

35.5GB available storage on macOS Sierra or later*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Big Sur:

MacBook (2015 or later)

MacBook Air (2013 or later)

MacBook Pro (Late 2013 or later)

Mac mini (2014 or later)

iMac (2014 or later)

iMac Pro (2017 or later)

Mac Pro (2013 or later)

Learn about compatibility and how to identify your specific Mac model

Don't use FireFox or WaterFox for this link. it'll mess it up...

https://apps.apple.com/us/app/macos-big-sur/id1526878132?ls=1&mt=12

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

Monterey…

https://apps.apple.com/us/app/macos-monterey/id1576738294

macOS Monterey on the App Store (apple.com)

Great news, take care my friend.:)

Rich_man wrote:

I hope this helps someone else in future.

Glad something worked for you; it would be helpful to others to know exactly what it was that worked....