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 $\mu(A)=\frac{n(A)}{|G|}$. 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 $\phi \psi(x)=\phi(x).\psi(x)$?

$$H=\bigm\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigm(\sum_g \phi(g)=0 \land Cov(X,Y)=0 \bigm ) \bigm \}$$

Remark 1: Evry character $\phi:G\to S^1=\mathbb{T}$ obviousely 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 the above $H$?

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 extention $H$ 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