Beforehand: I am not particularly algebraically educated and, especially, I do not have much background as far as free products of groups are concerned. So, it might well be that my question seems trivial (on the other hand, I could not find a quick answer to my question).
Let $(G_i)_{i=1,\ldots,n}$ be a family of copies of the cyclic group of order 2.
Then, consider the $n$-fold free product $$G := \ast_{i=1}^n G_i$$
What are the properties of $G$? In particular, is $G$ nilpotent? If yes, what is the nilpotency class of $G$? The only related theorem I am aware of is the Kurosh subgroup theorem, but I don't see a straightforward application to my case, esp. as I am looking for a central series, and therefore for normal subgroups of $G$.
For all $i$, $G'\cap G_i=1$ ($G'$ is the commutamt of $G$), since $G'$ consists of words of even lenght. By the Kurosh subgroup theorem $G'$ is free. So $G$ is an extension of a free group by an elementary Abelian $2$-group.