The fact that we can find such an O is true by the definition of the subset topology. This doesnt' answer the convex case tho.

(I had all the answers I wanted, so I'm okay with closing it. I'd do it myself if I knew how to do it.)

As I said above, I don't see where the linked paper give an example of an infinitely generated group with endomorphism group being $\mathbb{Z}$ (I'm probably just missing it), could someone share the page number ?

So one question remains (except if it's also answered in Asaf's post ':) ), can we, assuming choice holds, find another group than the finitely generated ones, satisfying $\mathrm{Aut}(G) = \mathbb{Z}/2\mathbb{Z}$ ?

@Wojowu Oh, yes, that does work I think ! I feel dumb for missing that, I had the answer right under my nose the whole time.

My question is kind of vague, I'm wondering what can we say about a group $G$ satisfying $Aut(G) \simeq \mathbb{Z}/2\mathbb{Z}$. In particular, because the same question with $Aut(G) \simeq 0$ depends on choice, I was wondering if here too, we needed choice to get a group that isn't one of the finitely generated ones. Your link answers this question negatively (so thanks !).

Indeed, I'll edit to remove them. As for the title, I'm precisely asking what happens when the group isn't finitely generated anymore, so I don't understand the proposed edit.