"An Internet connection is required to install macOS" error on MacBook Air

From your sideways image

This computer is try to reinstall macOS 15 Sequoia released in 2024

A computer can can run a 2024 version of macOS could never install OS X 10.12 Sierra related 9 years ago in 2016

So forget that idea

In the present condition of the computer it apples you are trying to Re-Install macOS 15 Sequoia and having some issue doing this

Suggest reading this article for Apple

If an error occurred while updating or installing macOS

Check your internet connection

The installer needs a stable internet connection to download macOS and get firmware and other information specific to your Mac model. Make sure that your Mac is connected to the internet.

Install from macOS Recovery

To ensure that no other software on your Mac can interfere with installation, install macOS from macOS Recovery.

For the same reason, if you’re installing via the App Store or Software Update instead of Recovery, it might help to start up in safe mode before installing.

Owl-53:

That looks more like Sonoma because the second letter doesn't have a horizontal component to it like an e would:

Don't know why the icon is Sequoia - maybe he was running Sequoia at the time.

Good catch 👍

Guess, Sierra is really Sonoma ?

The OP's installer icon is actually Sonoma:

Sequoia is this: