Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be an unramified extension with ring of integers $R$. Let $S \subset R$ be an order. What is the relationship between the residue field of $S$ and the residue field of $R$?
What I know: We can consider the example where $p = 7$, $K = \mathbb{Q}_7[x]/(x^2+1)$, $R = \mathbb{Z}_7[x]/(x^2+1)$, and $S = \mathbb{Z}_7[x]/(x^2 + 49)$. Then the residue field of $R$ is $\mathbb{F}_{49}$, but the residue field of $S$ is $\mathbb{F}_7$! So clearly the residue fields need not be the same, but I have no idea if anything more can be deduced!
The map from $S$ to $R$ certainly induces an inclusion of the residue fields of $S$ to $R$. (Any element of the residue field of $S$ lifts to a unit in $S$ which will still be a unit in $R$).
But that's all you can say; let $k' \subset k$ be any subfield of $k$, and let
$$\psi: R \rightarrow R/\pi_R = k$$
be the map to the residue field. Then you can take $S = \psi^{-1}(k')$.