Pseudometric spaces. From minimality to maximality in the groups of combinatorial self-similarities

The group of combinatorial self-similarities of a pseudometric space $(X, d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym} (X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{IP}$ the class of all pseudometric spaces $(X, d)$ for which every combinatorial self-similarity $\Phi\colon~X~\to~X$ satisfies the equality $d(x, \Phi(x))=0,$ but all permutations of metric reflection of $(X, d)$ are combinatorial self-similarities of this reflection. The structure of $\mathcal{IP}$ spaces is fully described.