Let $\cal F$ denote the group of all finitely-supported permutations of $\mathbb N$. Say that a finite subgroup $G$ of $\cal F$ is singular if $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ but no cyclic subgroup of $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ (this is equivalent to saying that some element in $G$ sends $1$ to $2$, another sends $1$ to $3$ but no element of $G$ has all of $1,2$ and $3$ in a single orbit).

The Klein group (products of disjoint transpositions on $\lbrace 1,2,3,4 \rbrace$) is in example of such a subgroup.

Question 1 : are there other simple examples of minimal singular subgroups ? Is there a parametric description of all of them up to isomorphism ?

Question 2 : Denote by ${\cal F}(i \to j)$ the set of all permutations in $\cal F$ sending $i$ to $j$. Say that a permutation $s\in {\cal F}(1 \to 2)$ and a permutation $t\in {\cal F}(1 \to 3)$ are related iff the subgroup generated by $s$ and $t$ is a minimal singular subgroup of $\cal F$. Given $s$, let $R(s)$ denoted the set of all $t$'s such that $s$ and $t$ are related. Does $R(s)$ admit a simple description ?

Of course, any answer to question 2 automatically provides an answer to question 1.