TITLE: Groups of order $64$ with abelian group of automorphism
QUESTION [7 upvotes]: 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:

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

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

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