In this article about the hypercenter of a profinite group, in the proof of the first theorem the author states that if $G$ is a profinite group, $M$ is an open normal subgroup of $G$ and $S$ is a Sylow subgroup of $G$ (which implies that $MS/M$ is a Sylow subgroup of $G/M$), then we have $$N_{G/M}\left(MS/M\right) = M\left(N_G\left(S\right)\right)/M.$$ It is easy to prove that $N_{G/M}\left(MS/M\right) = M\left(N_G\left(MS\right)\right)/M \geq M\left(N_G\left(S\right)\right)/M$ for any abstract group $G$, any $S \leq G$ and any $M \unlhd G$; but then I can't figure out the other inclusion.
Any help would be appreciated. Thanks in advance!