I recall having to reinstall the OS on a neighbour's Mac when the machine wouldn't boot (flashing question mark). In that case, the only option available was to use internet Recovery. Note that it's best to use an ethernet connection if possible.
- Pie Lover
Other macOS installation options (from https://support.apple.com/en-us/HT204904)
When you install macOS from Recovery, you get the current version of the most recently installed macOS, with some exceptions:
- On an Intel-based Mac, if you use Shift-Option-Command-R during startup, you're offered the macOS that came with your Mac, or the closest version still available. If you use Option-Command-R during startup, in most cases you're offered the latest macOS that is compatible with your Mac. Otherwise you're offered the macOS that came with your Mac, or the closest version still available.
See also Mac startup key combinations