If $M$ is the trivial module over ${\mathbb F}_2$ then, for $n \ge 4$, the dimension of $H^2(S_n,M)$ is $2$ and, for $n \ge 5$, the restriction $H^2(S_n,M) \to H^2(S_{n-1},M)$ is an isomorphism. So I think the answer to your question is yes for $n-2 \ge 5$, because non-split extensions of $M$ by $S_{n-2}$ restrict to nonsplit extensions of $M$ by $S_{n-3}$.

To justify those assertions, you could look at the presentations of the covering groups of $S_n$, which were computed by Schur originally. You can also find them, for example, in Huppert's book "Endliche Gruppen I", and you see from the presentations arising from the Coxeter presentation of $S_n$ that the covering groups of $S_n$ restrict to those of $S_{n-1}$ for $n \ge 5$.

The presentations of the extensions of $M$ by $S_n$ are as follows.

$\langle a_i\,(1 \le i \le n-1),\,t \mid t^2=[t,a_i]=1, a_i^2=x\, (1 \le i \le n-1),\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (a_ia_{i+1})^3=t\,(1\le i \le n-2),\,(a_ia_j)^2=y\, (4 \le i+2 \le j \le n) \rangle,$

where $x$ and $y$ are equal to $1$ or $t$, giving the four different isomorphism types of extensions (except that two of them are isomorphic when $n=6$).