Installing Mac OS X Lion

Start up from macOS Recovery

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac. Lok for Utilities menu in top toolbar.

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

APFS won't work for Sierra or earlier, Sierra or earlier needs MacOS Extended Journaled. High Sierra won't work for APFS on earlier Macs.

If all else fails...

https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/

Use http:// instead of https://

https://www.youtube.com/watch?v=ouNQS36dIBw&t=95s

The formula for caos: APFS+macOS High+EFI… - Apple Community

How to recover a Deleted Hard Drive durin… - Apple Community

That MBA should run Lion,but won't be of much use even if it does.

Start up from macOS Recovery

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Are you using a Lion compatible Mac and did you prepare 'Install Mac OS X Lion.app' from the downloaded InstallMacOSX.dmg? I just downloaded it from the site below and could install Lion although I prefer Snow Leopard or El Capitan in the supported Macs.

How to download and install macOS - Apple Support