Primitive $2^n$ centralizer group Is this true that for all positive integer $n$? 

There is no primitive $2^n$ centralizer group.

We know that, there exist $n$-centralizer groups for $n\neq 2, 3$, and there exists a primitive $n$-centralizer group for all odd $n\neq 3$.
 A: Here is a little background on what Babak is asking about:
A group G is said to be an n-centralizer group if { CG(z) : z in G } has exactly n elements.  A group is said to be a primitive n-centralizer group if both G and G/Z(G) are n-centralizer groups.  The very enjoyable paper Belcastro–Sherman (1994) shows that if G is 4-centralizer, then G/Z(G) is the Klein four-group.  Since the Klein four-group is 1-centralizer, there is no primitive 4-centralizer group.

Belcastro, Sarah Marie; Sherman, Gary J.
  Counting centralizers in finite groups.
  Math. Mag. 67 (1994), no. 5, 366–374.
  MR1307800
  DOI:10.2307/2690998

This is well suited to beginning group theory students.
However, the problem was taken up by other people, and in Ashrafi–Taeri (2005), conjecture 2.5 suggests that there is no primitive 2n-centralizer group.

Ashrafi, Ali Reza; Taeri, Bijan.
  Classification of finite groups by the number of element centralizers.
  Groups St. Andrews 2005. Vol. 1, 148–157, 
  London Math. Soc. Lecture Note Ser., 339,
  Cambridge Univ. Press, Cambridge, 2007.
  MR2327320 Google:books

I don't see much progress on this, other than the case 2n=8 settled by:

Abdollahi, A.; Jafarian Amiri, S. M.; Hassanabadi, A. Mohammadi.
  Groups with specific number of centralizers.
  Houston J. Math. 33 (2007), no. 1, 43–57.
  MR2287843
  Author's preprint.

Presumably Babak should write the authors and ask if they are aware of any further progress.  Ali Reza Ashrafi in particular wrote quite a few papers on this topic in the early part of last decade, and so probably has kept at least some contact with the problem.

In case you want to explore some examples in GAP, you might like to use (or improve) the functions I wrote to understand the problem:
centralizerCount := function(g)
  local inn;
  inn := Action( g, AsSet(g), OnPoints );
  return Size( AsSet( List( [1..Size(g)], i -> Stabilizer(inn,i) ) ) );
end;

gs := AllSmallGroups([1..60], g -> centralizerCount(g), 8 );;
gid := List(gs, IdGroup);
zid := List(gs, g -> IdGroup(g/Center(g)) );
counterexamples := Intersection( gid, zid );

Notice that the for any group G in the list 8-centralizer groups gs, the inner automorphism group G/Z(G) is no longer 8-centralizing.
