Let $G$ be a finite group, $k$ the field of complex numbers. 



> 
> Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$: 
>  $$\sigma(g,h)=\sigma(ghg^{-1},g)$$
> 
> 
> 


I would like to rule out nontrivial $\sigma$ with that property for any nonabelian group or get hands on a counterexample. They are directly related to possible additional automorphisms in nonabelian Dijkgraaf-Witten theory resp. additional braided autoequivalences of the Drinfel'd double representations.


A conjugacy class $[g]$ is called $\sigma$-*regular* iff $\sigma(g,h)=\sigma(h,g)$ for all $h\in Cent(g)$ (this does not depend on the representing $g$ or $\sigma$). Hence a symmetric cocycle in the sense above would especially imply that **all** conjugacy classes $[g]$ are $\sigma$-regular. We can hence reformulate the question:



> 
> Can a nontrivial $k\_\sigma[G]$ have the same number of irreducible representations as $k[G]$? 
> 
> 
> 


Thank you!


**Added:** The converse is also true! It seems standard (e.g. Ofir's reference [Higgs89] Lm. 1.2i) that in any class $[\sigma]$ there is a representative such that for any $\sigma$-regular $x$ and any $g$
$$\frac{\sigma(g,x)\sigma(gx,g^{-1})}{\sigma(g,g^{-1})}=1$$
An easy cohomology calculation shows indeed
$$\frac{\sigma(g,x)}{\sigma(gxg^{-1},g)}
=\frac{\sigma(g,x)}{\sigma(gxg^{-1},g)}
\cdot \frac{\sigma(gx,g^{-1})\sigma(gxg^{-1},g)}{\sigma(gx,1)\sigma(g,g^{-1})}=1$$
hance if all $x$ are $\sigma$-regular (second question) then this equation holds for all $g,x$ (first question). 



symmetric 2-cocycle / many projective representations
I will answer your second question.
By generalized Maschke theorem the twisted group algebras $\mathbb{C}^fG$ are all semi-simple and the number of simple components is the number of the irreducible projective $f$-representations of $G$, ($|Irr\_f(G)|$). Hence, 
$$|Irr\_f(G)|=dim (Z(\mathbb{C}^fG)).$$
Now, an element $g\in G$ is called $f$-regular if for any $x\in C\_G(g)$
$$f(g,x)=f(x,g).$$
It is also known that $Z(\mathbb{C}^fG)$ has a basis consisting of $f$-regular conjugacy classes. Clearly for trivial cocycle any conjugacy class is regular, hence I formulte your second question as follows:


Is there exist a group $G$ and a $f\in Z^2(G,\mathbb{C}^\*)$ which is not cohomogaclly trivial such that any $g\in G$ is $f$-regular.


This elements are sometimes called "distinguish" and they may exist as in the example gave by R.J.Higgs in his paper from 1987: "projective characters of degree one and the inflation-restriction sequence". 


