To generalize slightly the result mentioned by mjqxxxx, assume that one groups the original states $S_i$ into classes $C_a$. Hence the collection of classes $(C_a)$ is a partition of the state space and each class can be reduced to one state, or not. Then a sufficient condition for the resulting process to still be Markov is that the function $\varphi$ defined, for every state $S$ and class $C$, by the formula $$ \varphi(S,C)=\sum_{T\in C}P(S\to T) $$ depends on the state $S$ through its class only. In other words, one asks that, for every classes $C$ and $D$ and every states $S$ and $S'$ in $C$, $$ \varphi(S,D)=\varphi(S',D). $$ Note that the condition is always true if every class is reduced to one state, and that it is also true if there is only one class.
Caveat: if this condition is not met, it can nevertheless happen that the resulting process is Markov for some, but not all, initial conditions.