Quotients of ramification groups seen as additive subgroups of $S/Q$. I am working through Marcus's book $Number$ $Fields$, and I have been working for a while on exercise $22$ from chapter $4$. 
Letting $K \subset L$ be a Galios extions of a number field $K$, and let $S$ be the associated ring of integers of $L$. Then we can define $V_m(Q)$ be the $m$th ramification group of the prime ideal $Q$, meaning that 
$$V_m = \{\sigma \in Gal(L/K) : \sigma(\alpha) \equiv \alpha \mod Q^{m+1} \quad \forall \alpha \in S \}.$$
Now, it's fairly easy to see that $V_{m+1}(Q)$ is normal in $V_m(Q)$, and the exercise claims that this quotient group is a subgroup of the additive group of $S/Q$.
Fixing $\pi \in Q - Q^2$, the first step in the exercise is that if $\sigma \in V_{m-1}$ then show that there exists some $\alpha \in S$ so that $\sigma(\pi) \equiv \pi + \alpha \pi^m \mod Q^{m+1}$.
I've tried to set up some equivalence conditions using the Chinese remainder theorem, but was unable to make them work in the correct way. 
Any hints or suggestions on where to start would be fantastic. I feel pretty confident I can do the rest of the exercise if I can get this part.
 A: I think this is all the clues you need


*

*Because $S$ is integrally closed in $Frac(S)$, for any maximal ideal $Q$, with the localization $R = (S-Q)^{-1} S$, then $S/Q^{m+1} \cong R/(QR)^{m+1}$. 

*$QR$ is the unique maximal ideal of $R$ noetherian thus $QR = (\pi)$ for any $\pi \in QR, \pi \not \in (QR)^2$. Note if $Q$ isn't a principal ideal of $S$ then $\pi$ is an element of $Frac(S)$ not of $S$. Let $r = R \cap K$.

*For any set $B_m \subset R$ of representatives of $R/(\pi)^m$ then $R/(\pi)^{m+1} = \{ a+\pi b, a \in B_m, b \in B_1\}$.

*Thus $V_{m-1}/V_m$ is a subgroup of $H_m = \{ f \in Aut_{r-module}(R/(\pi)^{m+1}), f = Id \bmod (\pi)^m\}$
If $ f \in H_m$ then it is fully determined by $f(\pi)$ which must be of the form $ \pi+\alpha \pi^m$ for some $\alpha \in B_1$ ie. $\alpha \in R/(\pi)$. 

*The composition on elements $f_1,f_2 \in H_m$ gives a group law $f_1 \circ f_2(\pi)= (\pi+\alpha_2\pi^m)+\alpha_1 (\pi+\alpha_2\pi^m)^m=\pi +(\alpha_1+\alpha_2)\pi^m \bmod (\pi)^{m+1}$ ie.  $H_m \cong (R/(\pi),+)$ and $V_{m-1}/V_m$ is isomorphic to a subgroup of $(R/(\pi),+) \cong (S/Q,+)$
