I'm closer to understanding this thing now: it's a PUP. Potential unwanted program - and Malwarebytes (the free version I have) is the only thing that picked it up. I have three other products that could have, should have - but didn't.
It seems to be in my Google preferences, according to Malwarebytes (which reported the full name of the PUP) so I have just stopped using Chrome. Now, when I run Malwarebytes, it doesn't show up and those annoying messages don't show up - they are meant to make you think you need to click on a box to have the update installed.
It turns out Google Chrome wasn't the only thing it said needed updating, there were other products it 'reported.' Glad I hesitated and found out it's malware!
I'm using Firefox (I have other browsers) and I no longer have the problem.