Hello @Jean_Yves_Dhenain,
Let’s use the informal “tu” form with each other if that works for everyone (I had used the formal “vous” with Camille yesterday, following Zogstip’s lead). Ah, the French language 
Are you the “director” of Camille that she mentioned yesterday, or is there no connection? You seem to be on the same page as her. It’s hard for me to understand, I’ll admit (I’m more aligned with Benjamin).
Like him, and as I had tried to clarify with Camille yesterday, this notion of activity “outside the group” is incomprehensible to me and doesn’t seem to correspond to anything concrete (especially not technically at the forum level).
There is public and private activity. What is public is public to everyone, including members of a private group. So what they say publicly is visible to everyone, including other members of a private group they belong to. I don’t really see how one could think or want it otherwise. Or else, they should not say it publicly (via private message, or in a private category that might perhaps be part of another group, for example).
Imagine in real life a secret society whose members exchange information among themselves. If a member goes on TV and says something, everyone will potentially see it, including members of the secret society. How could it be otherwise? Now, if he is part of another secret society, only the members of that latter group will know what he says there (during the secret meetings of that group, of course). What exactly are you trying to achieve on your end? (possibly explaining it with an analogy from “real life”). A “private group” is not an invention of invisibility for its members, no matter what they do or where they do it.
EDIT: Okay, I think it just clicked Your “group activity” would only be what happens in a category dedicated to the group (potentially private). That is the technical part of what you and Camille apparently want to do. And then, you would simply want the group’s “activity” tab to display ONLY what happens in that category. Benjamin provided the solution to accomplish this above.
This does not mean that other members won’t see everything that is done publicly, or that they won’t be able to read it, but just that it won’t appear in the group’s “activity” tab (it will still appear in each member’s individual tab). Is that correct? (that is actually what Camille had explained. It is only a matter of presentation. Which I strangely had a hard time understanding)