Groups of order $64$ with abelian group of automorphism

Question

G. A. Miller in 1913 constructed the first example of a non-abelian group of order $64$ with abelian group of automorphisms. It is the group $$G=(C_8\rtimes C_4)\rtimes C_2=\langle x,y,z\colon x^8, y^4, z^2, yxy^{-1}=x^5, zxz^{-1}=x,zyz^{-1}=y^{-1}\rangle.$$ After few years, following observations were made:

1. There is no non-abelian group of order $<64$ with abelian group of automorphism.

2. There are (exactly) two more non-abelian group of order $64$ with abelian automorphism group.

Question: What are the other groups of order $64$ with abelian automorphism group? In the presentation, where they differs with $G$? (I mean, it may be a slight modification of $G$ above; if it is such, what is that modification?)

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Edit: James pointed error; there are two more non-abelian groups of order $64$, not one, with abelian automorphism group.

Answer 1

This doesn't seem to be quite correct. There appear to be three non-abelian groups of order $64$ with abelian automorphism group. They are:

SmallGroup( 64, 68 )
SmallGroup( 64, 69 )
SmallGroup( 64, 116 )

Miller's group is the last of these three.

EDIT: Assuming I haven't made any transcription errors here, a presentation for SmallGroup( 64, 68 ) is

$$\langle x,y,z \mid x^4, z^4, x^2 = y^2, z^2 = [y,x], [y,z], [x, z^2], [y, x^2], [z, x^2], [z,x] = [x^{-1},z] \rangle.$$ A presentation for SmallGroup( 64, 69 ) is:

$$\langle x,y,z \mid x^4, z^4, y^2 = z^2 x^2, z^2 = [y,x], [z,y], [z,x], [y,[z,x]], [x^2, z], [x^2, y], [z^2, x], [z,x] = [x, z^{-1}]\rangle.$$