Robust Monochromatic Cliques in a Random Edge-Coloring

Each edge of the complete graph $K_n$ is independently colored red or blue with equal probability $\frac{1}{2}$. For a fixed integer $k \ge 2$, find the expected number of monochromatic $k$-cliques (complete subgraphs on $k$ vertices whose edges are all the same color). Express your answer in terms of $n$ and $k$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.