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Note. Apparently for the computation of independent sets the graph whose vertices are right cosets in the subgroup $H$ and two vertices are connected by an edge can be useful if there is at least one edge between the elements of these cosets in the graph $AG_7$. I will try to draw this graph and post it here.

Note. Apparently for the computation of independent sets the graph whose vertices are right cosets of $H$ in $A_7$ and two vertices are connected by an edge can be useful if there is at least one edge between the elements of these cosets in the graph $AG_7$. I will try to draw this graph and post it here.

Addendum 3. (01.07.2022)

And this is the above graph with its $4$-vertice colouring.

R1:= [ (), (4,5,6), (4,6)(5,7), (3,4,6,7,5), (3,5,4,6,7)];;
R2:= [ (5,6,7), (4,5)(6,7), (4,7,6), (3,4,7,6,5) ];;
R3:= [ (5,7,6), (4,6,5), (4,7)(5,6) ];;
R4:= [ (4,5,7), (4,6,7), (3,4,5) ];;
S:=[(1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,2,7),(2,1,3),(2,1,4),(2,1,5),(2,1,6),(2,1,7)];;
q:=true;;
for a in Combinations(R1,2) do
    x:=a[1];
    y:=a[2];
    q:=IsEmpty(Intersection2(H, x*S*y^-1));
    if not q then break; fi;
od;
q;

R1:= [ (), (4,5,6), (4,6)(5,7), (3,4,6,7,5), (3,5,4,6,7)];;
R2:= [ (5,6,7), (4,5)(6,7), (4,7,6), (3,4,7,6,5) ];;
R3:= [ (5,7,6), (4,6,5), (4,7)(5,6) ];;
R4:= [ (4,5,7), (4,6,7), (3,4,5) ];;
S:=[(1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,2,7),(2,1,3),(2,1,4),(2,1,5),(2,1,6),(2,1,7)];;     
for a in Combinations(R1,2) do
    x:=a[1];
    y:=a[2];
    q:=IsEmpty(Intersection2(H, x*S*y^-1));
    if not q then break; fi;
od;
q;

Note. Apparently for the computation of independent sets the graph whose vertices are right cosets in the subgroup $H$ and two vertices are connected by an edge can be useful if there is at least one edge between the elements of these cosets in the graph $AG_7$. I will try to draw this graph and post it here.