Mac Osx Lion install

Your iMac late 2011 can only upgrade as far as macOS High Sierra 10.13.6

so don't try Mojave, Catalina or Big Sur.

Apple only provides its installers via download from the Apple Servers.

Anything you buy on the net is illegal and may not be trustworthy, expired certificates

malware etc.

If you know someone who has a mac of a similar vintage ask them to

download High Sierra via this link, they must use Safari to click on the link.

How to get old versions of macOS – Apple Support

Go to the Download OS section and click on macOS High Sierra

this will redirect you to the App Store High Sierra download it

should say Get or Download.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS High Sierra.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

The Install macOS High Sierra.app downloads to the Applications folder should be 5.23GBs in size.

Create a bootable USB installer using the instructions here,

How to create a bootable installer for macOS – Apple Support

Once you have done that then you can insert the bootable USB

directly into your mac, boot to it and then install High Sierra.

The certificates for several of Apples OS's expired in October 2019, Apple have been very

poor at getting these updated on their Recovery Servers with valid certificates.

This may be the problem you are experiencing. 

Try this workaround, we are going to set the time and date on your mac to a date previous to the expiry date of the certificates.

Connect your mac to your router via cable, not WiFi

Boot to your Recovery HD, click on Utilities in the menubar select Terminal.

Make sure WiFi is switched off, it can reset the date back to today.

Enter a new date, for example or just copy and paste

sudo date -u 011421002015

press Return

enter your password

press Return

If Terminal returns an error saying sudo : command not found, then try again without sudo.

Now try downloading the OS.

If this works then when the OS is installed and booted up you can Open System Preferences> Date & Time

and reset the time back to today.

Did you reset the date using Terminal making sure the WiFi stays off.

Apple does seem to be making the installation of older OS's increasingly difficult.

What mac is it, what year is it, you may be able to install a more recent OS.

Do you have access to another mac that you could use to download an OS installer app

then create a bootable USB installer so you can install to your original mac.