Non-normal Klein four-subgroups of symmetric group:S4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) Klein four-group and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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We consider the subgroup $H$ in the group $G$ defined as follows.

$G$ is the symmetric group of degree four, which for concreteness we take as the symmetric group on the set $\{1,2,3,4\}$.

$H$ is the Young subgroup for the partition $\{\{1,2\},\{3,4\}\}$. Explicitly, it is the subgroup comprising those permutations that send each of the subsets $\{1,2\}$ and $\{3,4\}$ to within itself. $H$ has a total of three conjugates, listed below:

Our local name	Partition stabilized	Set of all elements in the stabilizing subgroup
$H$	$\{\{1,2\},\{3,4\}\}$	$\{(),(1,2),(3,4),(1,2)(3,4)\}$
$H_{1}$	$\{\{1,3\},\{2,4\}\}$	$\{(),(1,3),(2,4),(1,3)(2,4)\}$
$H_{2}$	$\{\{1,4\},\{2,3\}\}$	$\{(),(1,4),(2,3),(1,4)(2,3)\}$

$H$ (and hence each of its conjugate subgroups) is isomorphic to the Klein four-group. However, $G$ has another subgroup isomorphic to the Klein four-group that is not one of these conjugate subgroups, and is not automorphic to these either. That is the normal Klein four-subgroup of symmetric group:S4 that comprises the identity element and the three double transpositions. The current article is not about that subgroup.