The best way to have buttons for specific languanges in Buttonizer is to create a group for each language.
You can simply use Duplicate to duplicate your main group for this.
Then you can use Page Rules to make the groups only appear on their respective languages.
Here's a video on how to set it up for the 3 languages that you specified:
Setting up page rules
To explain the Page rules:
For the main language, Italian, the conditions consists of the other languages.
We then make the Italian group hide when the Page rule is triggered, which in this case, it's when the user is on a page of the other languages.
For the other languages, English and French, the conditions only have to specify their respective language codes.
We then make their groups only show when the Page rule is triggered.
Let me know if you were able to set this up!