$\DeclareMathOperator\Cov{Cov}$

Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X,Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

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.