When I had a problem downloading the Mac OS to my iMac , I thought I would download it on an external hard drive to make it work. but unfortunately after the software was downloaded I found out that the operating system overlapped my hard dive and formatted it, and it is APFS format. how can I restore it ??
iMac 24″, macOS 13.2
If that drive originally used the MacOS Extended file system (aka HFS+), then the macOS installer would have automatically converted the file system to APFS, but any previous data should still be on the drive although it will have been relocated to the "/Users/Shared/Relocated Items" folder since macOS 10.15+ do not allow anything to reside on the root of a macOS boot drive anymore.
About the read-only system volume in macOS Catalina or later - Apple Support
If, however, you actually erased that drive using Disk Utility, then those previous files are now gone....overwritten by the new OS.
I don't need an operating system anymore as I bought a new iMac . My problem is that I need my data back from the hard drive that was formatted. how can I get my data back and restore the hard drive ?
What OS version???
10.15 & above demand APFS to install, MacOS Extended Journaled is used by 10.12 & below... 10.13 & 10.14 can be forced to use either Format.
I don't remember what version.
but my problem is that the external hard drive that I used to install the operating system is formatted. how can I recover the data ?
Sounds like you need another external drive, formatted to APFS & install a minimum OS of 10.13.6
Well having an OS on an external drive is one way, another would be Target Disk Mode possibly...
Transfer files between two Mac computers using target disk mode - Apple Support
My external hard drive was formatted after I downloaded Mac OS software to it.
