= Solution
The <Markov property> and conditioning on the intermediate state give, for the full history through time $2n$,
$$
P(X_{2n+2}=j\mid X_0,\ldots,X_{2n})=(P^2)_{X_{2n},j}.
$$
Condition this identity again on the smaller history $(Y_0,\ldots,Y_n)$. Its right-hand side is already determined by $Y_n=X_{2n}$, so the tower property gives
$$
\boxed{P(Y_{n+1}=j\mid Y_0,\ldots,Y_n)=(P^2)_{Y_n,j}.}
$$
Thus the sampled sequence is a time-homogeneous <Markov chain> with <transition matrix> $P^2$. Checking only its one-time marginal distributions would not by itself establish the <Markov property>.
It is the every-second-step case of a <subsampled Markov chain>.
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