Orbit augmentation kernel and rank formula

For every field $k$, every group $G$, and every finite type $X$ equipped with a left $G$-action and a finite enumeration, let $A\subseteq k^{(X)}$ be the span of $\delta_{g\cdot x}-\delta_x$ over all $g\in G$ and $x\in X$, where $k^{(X)}$ is the vector space of finitely supported functions and $\delta_x$ has value $1$ at $x$ and $0$ elsewhere. Let $X/G$ be the quotient of $X$ by the equivalence relation $x\sim y$ if $x=g\cdot y$ for some $g\in G$, and let $q_*:k^{(X)}\to k^{(X/G)}$ send each $\delta_x$ to $\delta_{[x]}$, so that the coefficient of an orbit in $q_*f$ is the sum of the coefficients of $f$ over that orbit. The assertion is the conjunction $A=\ker(q_*)$ and $\dim_k A=|X|-|X/G|$, where the right-hand subtraction is natural-number subtraction, truncated at zero, and $|X/G|$ denotes the finite cardinality of the orbit quotient. The group $G$ may be infinite, the field has arbitrary characteristic, and $X$ may be empty, in which case both subspaces and both cardinalities are zero.