$\DeclareMathOperator\Cov{Cov}$Assume that  $G$  is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm.
We consider $G$ as a sample (probability)  space  with the normalized counting measure  $\mu(A)=\frac{|A|}{|G|}$.  So we consider  $X$, $Y$  as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

>Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$?
> $$H=\bigl\{\phi=X+iY:G\to S^1  \bigm| ( \phi\equiv1) \lor  \bigl(\sum_g \phi(g)=0 \land  \Cov(X,Y)=0 \bigr ) \bigr \}.$$


**Remark 1:** Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

**Remark 2:** What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?



**Edit** After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the  following. In fact the implicit and initial goal of this post was:

> Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelated" (or/and) "zero-mean property".

Please see this related post:
https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.