Let $G$ be a group with finite subsets $A,B \subseteq G$. Let $k$ be an integer between $1$ and $|A|$ (or |A|/2 if it helps), and let $C$ be a random subset of $A$ of size $k$, chosen uniformly out of all such sets. We take $\mu_C = \frac{|A|}{k}1_C$ (where $1_X$ is an indicator function for a set $X$).
For $f,g : G \rightarrow \mathbb{C}$, we have the convolution $f*g(x) = \sum_{y \in G}f(y)g(y^{-1}x)$.
I was able to show that $\mathbb{E}\left[\mu_C * 1_B\right] = 1_A * 1_B(x)$, but it's also supposed to be "easy to see" that $\operatorname{Var}(\mu_C * 1_B(x)) \leq \frac{|A|}{k} 1_A*1_B(x)$. The bounds for $\operatorname{Var}(\mu_C * 1_B(x))$ that I can come up with all seem to be off by at least an extra factor of $\frac{|A|}{k}$.
Any help is appreciated, thanks!