The Markov property can fail for a larger filtration

Let $(X_n)$ be IID fair bits and [F1] ${\mathcal{F}}_n^X=\sigma (X_0,\dots ,X_n)$. By [F1], $X$ is a Markov chain for this filtration with $K(x,\{1\})=1\mathrm{/}2$.

Define a larger filtration by [F2] ${\mathcal{G}}_0=\sigma (X_0,X_1),{\mathcal{G}}_n=\sigma (X_0,\dots ,X_n,X_1)(n\ge 1)\mathrm{.}$ Then ${\mathcal{G}}_0={\mathcal{G}}_1$ and ${\mathcal{G}}_n={\mathcal{F}}_n^X$ for $n\ge 1$, so $({\mathcal{G}}_n)$ is increasing; it contains ${\mathcal{F}}_n^X$ at every time, and $X$ is adapted. Thus it meets the structural requirements in [F2].

Since $X_1$ is ${\mathcal{G}}_0$-measurable, [F2, step 1.1, step 1.2] $\mathbb{P}(X_1=1\mid {\mathcal{G}}_0)=1_{\{X_1=1\}}$ almost surely. This differs from $K(X_0,\{1\})=1\mathrm{/}2$ on both positive-probability events $\{X_1=0\}$ and $\{X_1=1\}$. Hence the time-zero identity in [F2] fails for the larger filtration, although it holds naturally. Empty/full target events still give zero/one and do not witness failure. Choice is used only by the conditional-probability classes.