co.combinatorics - Transitivity-related property of finite permutation groups

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 $sin {cal F}(1 to 2)$
and a permutation $tin {cal F}(1 to 3)$ 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.