  We answer a question of A. Skalski and P.M. So{\l}tan (2016) about inner
faithfulness of the S.~Curran's map of extending a quantum increasing sequence
to a quantum permutation in full generality. To do so, we exploit some novel
techniques introduced by Banica (2018) and Brannan, Chirvasitu, Freslon (2018)
concerned with the Banica's conjecture regarding quantum permutation groups.
Roughly speaking, we find a inductive setting in which the inner faithfulness
of Curran's map can be boiled down to inner faithfulness of similar map for
smaller algebras and then rely on inductive generation result for quantum
permutation groups of Brannan, Chirvasitu and Freslon.
