Thanks for all your responses. The weirder thing is that when I went back to my laptop the next day, the issue appeared to be gone. I haven't seen any Sophus notifications since, which is also suspicious.
Yes, it's my personal machine. I do recall that as I was working on something earlier that evening, there was something that flashed up on the screen and disappeared before I could focus on it and see what it was, because I was already in the process of clicking on something when it popped up, so it was only on the screen for a nanosecond before I clicked on the button. Does that make sense? I was actually reaching for the button to click, and as I clicked it, this "thing" popped up. But I have no idea what it was because it disappeared as soon as I clicked the button (and saw it simultaneously). So that could be how it got installed?
varenchris, I had the same concerns about it not really being Sophos, too. As I said, I was able to download the real Sophos installer, but Apple wouldn't allow it to run because it couldn't run a malware check on it.
The other factor that may come into play is that particular evening, I had booted from an external hard drive to work on my taxes (which I hadn't done for quite some time). I use Quicken Home & Business, which doesn't have a Mac version, so I have Parallels and Windows installed on the external drive which is actually the hard drive from my previous Mac, along with Quicken, and I boot to that drive to work on taxes (because I don't want Windows on this machine). So maybe I need to boot to the external drive in safe mode and see if that's where the problem is? But the Sophos popups occurred after I shut down and disconnected the external drive and re-booted to the internal hard drive, so I don't see how that would happen if the problem is on the external drive.
At one time, I had PCMatic installed on this machine (yes, it works on Mac, too), but the guys in my local computer shop told me once to uninstall it because it doesn't "play nice" with Mac and could be causing whatever issue I was having at the time.
Any other thoughts?
Thanks again for your responses.