Banach-Stone-like results for combinatorial Banach spaces

We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $\kappa$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $\kappa$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $\omega$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.