Derived subgroup of nontrivial semidirect product of Z4 and Z4

Definition
Consider the group:

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

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:

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

This is the derived subgroup. In particular, it is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.

Cosets
The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. The subgroup has order 2 and index 8, so there are 8 cosets, given as:

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

Complements
The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

GAP implementation
The group and subgroup can be constructed as follows, using the SmallGroup and DerivedSubgroup functions:

G := SmallGroup(16,4); H := DerivedSubgroup(G);

Implementing this in GAP looks as follows:

gap> G := SmallGroup(16,4); H := DerivedSubgroup(G);  Group([ f3 ])

Below is GAP implementation to test some of the assertions in this page:

gap> StructureDescription(G/H); "C4 x C2" gap> Order(G); 16 gap> Order(H); 2 gap> Index(G,H); 8 gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true