Some computations following the answer of John Shareshian, pointing out one interesting anomaly:

For $G=A_8$ and the partitions $\pi_1=\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}$, $\pi_2=\{\{1,2,3,4\},\{5,6,7,8\}\}$.

```
gap> G:=AlternatingGroup(8);
Alt( [ 1 .. 8 ] )
gap> P1:=[[1,2],[3,4],[5,6],[7,8]];
[ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ 7, 8 ] ]
gap> P2:=[[1,2,3,4],[5,6,7,8]];
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ] ]
gap> G1:=Stabilizer(G,P1,OnSetsDisjointSets);
Group([ (5,6)(7,8), (3,4)(7,8), (1,2)(7,8), (3,4)(5,6), (3,6,4,5)(7,8), (3,5,8)(4,6,7), (1,3,5,8)(2,4,6,7) ])
gap> G2:=Stabilizer(G,P2,OnSetsDisjointSets);
Group([ (5,8,7), (5,8)(6,7), (5,7)(6,8), (3,4)(7,8), (2,3,4), (1,4)(2,3), (1,3)(2,4), (1,5,4,7,3,6)(2,8) ])
```

Note that $G_1$ is **NOT** a maximal subgroup of $G$ (whereas $G_2$ is):

```
gap> IntermediateSubgroups(G,G1);
rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 1, 3 ], [ 2, 3 ] ], subgroups := [ Group([ (1,8)(2,7), (1,8,3)(2,7,4), (1,5,2,6)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(7,8), (3,4)(7,8), (5,6)(7,8), (2,3)(6,7) ]), Group([ (1,8)(2,7), (1,8,3)(2,7,4), (1,5,2,6)(3,8,4,7), (1,4,2,3)(5,7,6,8), (1,2)(7,8), (3,4)(7,8), (5,6)(7,8), (2,3)(6,8) ]) ] )
gap> IntermediateSubgroups(G,G2);
rec( inclusions := [ [ 0, 1 ] ], subgroups := [ ] )
```

Finally, $[H,G]$ is Boolean of rank $3$ (and not $2$ as expected) (it is exactly the example given in the question):

```
gap> H:=Intersection(G1,G2);
Group([ (5,8)(6,7), (5,7)(6,8), (3,4)(5,7,6,8), (1,2)(5,7,6,8), (1,3)(2,4)(5,8)(6,7), (1,5,4,8)(2,6,3,7) ])
gap> IntermediateSubgroups(G,H);
rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 4 ], [ 2, 6 ], [ 3, 5 ], [ 3, 6 ], [ 4, 7 ], [ 5, 7 ], [ 6, 7 ] ],
subgroups := [ Group([ (1,5)(2,6)(3,7)(4,8), (3,4)(7,8), (2,3,4)(6,7,8), (1,4)(2,3), (5,8)(6,7), (5,7)(6,8), (1,3)(2,4) ]), Group([ (1,5)(2,6)(3,7)(4,8), (3,4)(7,8), (2,4,3)(6,7,8), (1,4)(2,3), (5,8)(6,7), (5,7)(6,8), (1,3)(2,4) ]), Group([ (3,5,8)(4,6,7), (3,4)(5,7,6,8), (1,3)(2,4)(5,7)(6,8) ]), Group([ (1,3)(2,4), (2,3,4), (5,7)(6,8), (1,2,3,4)(5,8,7,6), (1,2,3,4)(5,6), (1,5)(2,6)(3,7)(4,8) ]), Group([ (5,7)(6,8), (2,3,5)(4,7,6), (1,2)(3,4)(5,6)(7,8) ]), Group([ (5,7)(6,8), (1,3,5)(4,7,6), (1,2)(3,4)(5,6)(7,8) ]) ] )
```

Note that this anomaly does not appear for $G=S_8$:

```
gap> G:=SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> G1:=Stabilizer(G,P1,OnSetsDisjointSets);
Group([ (7,8), (5,6), (3,4), (1,2), (3,5,4,6), (3,5,8)(4,6,7), (1,3,5,8)(2,4,6,7) ])
gap> G2:=Stabilizer(G,P2,OnSetsDisjointSets);
Group([ (7,8), (5,8,7), (5,8)(6,7), (5,7)(6,8), (3,4), (2,3,4), (1,4)(2,3), (1,3)(2,4), (1,5,4,7,3,6)(2,8) ])
gap> H:=Intersection(G1,G2);
Group([ (5,6), (5,7,6,8), (3,4)(5,7,6,8), (1,2)(5,7,6,8), (1,3)(2,4)(5,7,6,8), (1,5,3,7,2,6,4,8) ])
gap> IntermediateSubgroups(G,H);
rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 1, 3 ], [ 2, 3 ] ],
subgroups := [ Group([ (1,2), (3,4), (5,6), (7,8), (1,3,5,7)(2,4,6,8), (1,3)(2,4) ]), Group([ (1,2,3,4), (1,2), (5,6,7,8), (5,6), (1,5)(2,6)(3,7)(4,8) ]) ] )
```

Now, for $G=A_{16}$, and the partitions $\pi_1=\{\{1,2\},\{3,4\},\{5,6\},\{7,8\},\{9,10\},\{11,12\},\{13,14\},\{15,16\}\}$, $\pi_2=\{\{1,2,3,4\},\{5,6,7,8\},\{9,10,11,12\},\{13,14,15,16\}\}$, $\pi_3=\{\{1,2,3,4,5,6,7,8\},\{9,10,11,12,13,14,15,16\}\}$

```
gap> G:=AlternatingGroup(16);
Alt( [ 1 .. 16 ] )
gap> P1:=[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]];
[ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ 7, 8 ], [ 9, 10 ], [ 11, 12 ], [ 13, 14 ], [ 15, 16 ] ]
gap> P2:=[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]];
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ], [ 13, 14, 15, 16 ] ]
gap> P3:=[[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]];
[ [ 1, 2, 3, 4, 5, 6, 7, 8 ], [ 9, 10, 11, 12, 13, 14, 15, 16 ] ]
gap> G1:=Stabilizer(G,P1,OnSetsDisjointSets);
<permutation group of size 5160960 with 17 generators>
gap> G2:=Stabilizer(G,P2,OnSetsDisjointSets);
<permutation group of size 3981312 with 19 generators>
gap> G3:=Stabilizer(G,P3,OnSetsDisjointSets);
<permutation group of size 1625702400 with 18 generators>
gap> H:=Intersection(Intersection(G1,G2),G3);
<permutation group with 12 generators>
gap> IntermediateSubgroups(G,H);
rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 5 ], [ 2, 6 ], [ 3, 4 ], [ 3, 6 ],[ 4, 7 ], [ 5, 7 ], [ 6, 7 ] ], subgroups := [ Group([ (1,15,4,14)(2,16,3,13), (1,10,8,14)(2,9,7,13)(3,12,6,16)(4,11,5,15), (7,8)(13,14) ]), Group([ (1,9,8,16,6,13)(2,10,7,15,5,14)(3,12)(4,11), (3,7)(4,8), (7,8)(11,12) ]),
<permutation group of size 1327104 with 18 generators>, <permutation group of size 3981312 with 13 generators>,
<permutation group of size 5160960 with 9 generators>, <permutation group of size 1625702400 with 6 generators>
] )
```

There is no anomaly, we get a Boolean lattice of rank $3$. Idem for $G=S_{16}$:

```
gap> G:=SymmetricGroup(16);
Sym( [ 1 .. 16 ] )
gap> G1:=Stabilizer(G,P1,OnSetsDisjointSets);
<permutation group of size 10321920 with 15 generators>
gap> G2:=Stabilizer(G,P2,OnSetsDisjointSets);
<permutation group of size 7962624 with 17 generators>
gap> G3:=Stabilizer(G,P3,OnSetsDisjointSets);
<permutation group of size 3251404800 with 17 generators>
gap> H:=Intersection(Intersection(G1,G2),G3);
<permutation group with 12 generators>
gap> IntermediateSubgroups(G,H);
rec( inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 5 ], [ 2, 6 ], [ 3, 4 ], [ 3, 6 ], [ 4, 7 ], [ 5, 7 ], [ 6, 7 ] ],
subgroups := [ Group([ (1,13,3,15)(2,14,4,16), (1,13,11,5)(2,14,12,6)(3,15,9,7)(4,16,10,8), (13,14) ]), Group([ (1,13,5,11,3,15)(2,14,6,12,4,16)(7,9)(8,10), (3,7)(4,8), (15,16) ]),
<permutation group of size 2654208 with 19 generators>, <permutation group of size 7962624 with 10 generators>, <permutation group of size 10321920 with 10 generators>,
Group([ (1,2,3,4,5,6,7,8), (1,2), (9,10,11,12,13,14,15,16), (9,10), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16) ]) ] )
```

I tried to go further with $G=A_{32}$ (or $S_{32}$):

```
gap> G:=AlternatingGroup(32); #or SymmetricGroup(32);
Alt( [ 1 .. 32 ] )
gap> P1:=[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]];
[ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ 7, 8 ], [ 9, 10 ], [ 11, 12 ], [ 13, 14 ], [ 15, 16 ], [ 17, 18 ], [ 19, 20 ], [ 21, 22 ], [ 23, 24 ], [ 25, 26 ], [ 27, 28 ], [ 29, 30 ], [ 31, 32 ] ]
gap> P2:=[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24],[25,26,27,28],[29,30,31,32]];
[ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8 ], [ 9, 10, 11, 12 ], [ 13, 14, 15, 16 ], [ 17, 18, 19, 20 ], [ 21, 22, 23, 24 ], [ 25, 26, 27, 28 ], [ 29, 30, 31, 32 ] ]
gap> P3:=[[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16],[17,18,19,20,21,22,23,24],[25,26,27,28,29,30,31,32]];
[ [ 1, 2, 3, 4, 5, 6, 7, 8 ], [ 9, 10, 11, 12, 13, 14, 15, 16 ], [ 17, 18, 19, 20, 21, 22, 23, 24 ], [ 25, 26, 27, 28, 29, 30, 31, 32 ] ]
gap> P4:=[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]];
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ], [ 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 ] ]
gap> G1:=Stabilizer(G,P1,OnSetsDisjointSets);
<permutation group of size 685597979049984000 with 37 generators>
gap> G2:=Stabilizer(G,P2,OnSetsDisjointSets);
<permutation group of size 2219118333788160 with 37 generators>
gap> G3:=Stabilizer(G,P3,OnSetsDisjointSets);
<permutation group of size 31714899520389120000 with 35 generators>
gap> G4:=Stabilizer(G,P4,OnSetsDisjointSets);
<permutation group of size 437763136697395052544000000 with 34 generators>
gap> H:=Intersection(Intersection(Intersection(G1,G2),G3),G4);
<permutation group with 22 generators>
```

but on my laptop, GAP *shuts down* on the following computation after 2min30s...

```
gap> IntermediateSubgroups(G,H);
```

If someone can make this computation, I would be very interested in the result.

*Remark*: For $G=A_{2^n}$ (resp. $S_{2^n}$) and the equipartitions and $H$ as above, we observe that $|H| = 2^{2^n-2}$ (resp. $2^{2^n-1}$) for $n=2,3,4,5$. Is it true in general? In addition, for $n=2,3$, we observe that $H$ is the unique subgroup of such order. Is it true in general?