  Given a finite crystallographic root system $\Phi$ whose Dynkin diagram has a
non-trivial automorphism, it yields a new root system $\Phi_{\tau}$ by a
so-called classical folding. On the other hand, Lusztig's folding (1983) folds
the root system of type $E_8$ to $H_4$ starting from an automorphism of the
root lattice of type $E_8.$ The notion of a twisted quadratic folding of a root
system was introduced by Lanini-Zainoulline (2018) to describe both the
classical foldings and Lusztig's folding on the same footing. The structure
algebra $\mathcal{Z}(\mathcal{G})$ of the moment graph $\mathcal{G}$ associated
with a finite root system and its reflection group $W$ is an algebra over a
certain polynomial ring $\mathcal{S},$ whose underlying module is free with a
distinguished basis $\{\sigma^{(w)} \mid w \in W\}$ called combinatorial
Schubert classes. By Lanini-Zainoulline (2018), a twisted quadratic folding
$\Phi \rightsquigarrow \Phi_{\tau}$ induces an embedding of the respective
Coxeter groups $\varepsilon: W_{\tau} \hookrightarrow W$ and a ring
homomorphism $\varepsilon^*: \mathcal{Z}(\mathcal{G}) \rightarrow
\mathcal{Z}(\mathcal{G}_{\tau})$ between the corresponding structure algebras.
This paper studies the $\varepsilon^*$-preimage of Schubert classes and
provides a combinatorial criterion for a Schubert class $\sigma^{(u)}_{\tau}$
of $\mathcal{Z}(\mathcal{G}_{\tau})$ to admit a Schubert class $\sigma^{(w)}$
of $\mathcal{Z}(\mathcal{G})$ such that the relation
$\varepsilon^*(\sigma^{(w)}) = c \cdot \sigma^{(u)}_{\tau}$ holds for some
nonzero scalar $c.$
