Does avast install Google chrome?
It might well offer it during (or after) installation, the problem being it is pre-checked and if you aren’t alert you could well get it…
Do a custom installation so things don’t happen automatically (fast) and you can also choose the components that you need/want.
As I remember you can decide during installation if you want that. It is still an very unsual behaviour to press customers to use unwanted external software! 
Whilst I agree with the sentiment expressed, I actually liked having the SafeBrowser as an option to test some sites I was unsure about. However Chrome dropped support for XP a while back, meaning I am stuck with insecure browsers for the moment.
Opera 12(?) will still work and although a oldie, it is pretty secure considering the technology from back then.