homomorphism of profinite groups

Question

Let $G$ be a profinite group and consider a continuous surjective homomorphism: $$\phi:G\rightarrow \widehat{\mathbb Z}$$ where $\widehat{\mathbb Z}:=\varprojlim \mathbb Z/n\mathbb Z$. Moreover Let $H<G$ be a (closed) subgroups such that $f=|\widehat{\mathbb Z}:\phi(H)|<\infty$. I Have two questions:

1. Why the map $\frac{1}{f}\phi:H\rightarrow\widehat{\mathbb Z}$ is well defined? I mean: why if $a\in\phi(H)$ then $\frac{1}{f}a\in \widehat{\mathbb Z}$? $f$ is a natural number, so its inverse is not supposed to be in $\widehat {\mathbb Z}$.
2. Why $\frac{1}{f}\phi:H\rightarrow\widehat{\mathbb Z}$ is surjective?

Answer 1

If $\phi(H)$ has index $f$ in $\widehat{\mathbb{Z}}$, then $\phi(H)$ must in fact be the subgroup $f \widehat{\mathbb{Z}}$, since it is the unique subgroup with that index. This answers both of your questions.