I'm little bit puzzled with the following question, I was wondering if someone could help with a suggestion or a hint.

Let start with some notations. Suppose that $G$ is a Hausdorff topological group such that

- As a space $G$ is contractible.
- As a group $G$ is isomorphic to the free product $\mathrm{Q}\ast F(X) $ where $\mathrm{Q}$ is the additive abelian group of rational numbers and $F(X)$ is a free group generated by a set $X$.
- We assume that $\mathrm{Q}$ is a topological subgroup of $G$ in an obvious way and that $H=\{1\}\ast_{\mathrm{Q}} G$ is a retract of a free topological group (generated by a CW-complex).

Let $\mathbf{Ab}: \mathsf{Tgr}\rightarrow \mathsf{AbTgr}$ be the abelianization functor from the category of topological groups to abelian topological groups.

Since $G$ is a contractible group, for any natural number $n>0$ the power maps $p^{n}: G\rightarrow G$, $t\mapsto t^n$ are weak homotopy equivalences of underlying topological spaces.

My question is the following: is there a hope that the power maps: $p^{n}: \mathbf{Ab}(G)\rightarrow \mathbf{Ab}(G)$ induce weak homotopy equivalences of underlying topological spaces? The same question of $H$ and power maps $p^{n}: H\rightarrow H$, are they weak homotopy equivalences ?

Thank you in advance for any help!