Robust Law of Total Variance with a Random Number of Coin Flips

You first draw $N \sim \operatorname{Poisson}(5)$. Then, given $N = n$, you flip a fair coin $n$ times and let $X$ be the number of heads. What is $\operatorname{Var}(X)$?

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