Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4

Definition
Consider the group (here $$e$$ denotes the identity element):

$$G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$$.

This is a group of order 16 with elements:

$$\! \{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}$$

We are interested in the subgroup:

$$\! H = \{ e, y^2 \}$$

This is a subgroup generated by the square $$y^2$$ which is not a commutator (the only commutators are $$e$$ and $$x^2$$).

The quotient group $$G/H$$ is isomorphic to dihedral group:D8. We can see this by noting that the presentation of $$G/H$$ is the same as that of $$G$$ but now with the additional constraint that $$y^2$$ is the identity element. This is precisely the presentation of the dihedral group of order 8.

Cosets
The subgroup is a normal subgroup, so the left and right cosets coincide. The 8 cosets are:

$$\! \{ e, y^2 \}, \{ x, xy^2 \}, \{ x^2, x^2y^2\}, \{ x^3, x^3y^2 \}, \{ y, y^3 \}, \{ xy, xy^3 \}, \{ x^2y, x^2y^3 \}, \{ x^3y, x^3y^3 \}$$

GAP implementation
The group and subgroup pair can be constructed as follows:

G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G))));

Here is the GAP display for this:

gap> G := SmallGroup(16,4); H := Group(Difference(Set(List(G,x -> x^2)),Set(DerivedSubgroup(G))));  

Here is GAP code to verify some of the assertions in this page:

gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> StructureDescription(H); "C2" gap> StructureDescription(G/H); "D8" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); false