Automate a different website via my website

First of all, sorry if I make mistakes in my English language.

I have a website on Linux Ubuntu, I just want to automate different website, but users can see whats automate on different in browser's second tab. The exaple scenario is ;

User click a button in my website. This click is open a second tab in browser to -for example- google.com, write someting and click to search button automatically. User do not anything excluding click button in my website.

I am search a lot of thing on the internet for that. There are Selenium, puppeteer, Laravel Dock etc. But they for headless browser or just one user that having code on their pc. I want to everybody can do that which click button on my website.

How can I do that?