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 $\phi \psi(x)=\phi(x).\psi(x)$?

$$H=\{\phi=X+iY:G\to S^1\mid \text{either}\quad \phi=1 or \sum_g \phi(g)=0\quad \text{and}\quad Cov(X,Y)=0\}$$

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 $H$?