Write $S_3$ for the symmetric group on 3 letters.
The question: What are the possible extensions of $S_3$ by $\mathbb{Z}$ (up to equivalence)? (To avoid ambiguity, by an extension of G'' by G' I mean a short exact sequence of groups 1 \to G' \to G \to G'' \to 1.)
It is well-known that these are enumerated by the second cohomology group of $S_3$ with $\mathbb{Z}$-coefficients. There are two $S_3$-module structures on $\mathbb{Z}$. I do know that $H^2(S_3;\mathbb{Z}) = \mathbb{Z}/2$; here $\mathbb{Z}$ denotes the trivial $S_3$-module. I am, however, unsure about $H^2(S_3; \tilde{\mathbb{Z}})$ -- what I came up thus far is that it is either $\mathbb{Z}/2$ or $\mathbb{Z}/6$. EDIT: Actually, $H^2(S_3;\tilde{\mathbb{Z}}) = \mathbb{Z}/3$; I did my computations wrong previously.
I can think of only three extensions of $S_3$ by $\mathbb{Z}$: the obvious ones, $\mathbb{Z} \times S_3$ and $\mathbb{Z} \rtimes S_3$, and the infinite dihedral group $\mathbb{Z} \rtimes \mathbb{Z}/2$. EDIT: The fourth one is $\mathbb{Z}/3 \rtimes \mathbb{Z}$, which leaves me with only one group missing. Any ideas?