  We introduce the notion of $\imath$Schur superalgebra, which can be regarded
as a type B/C counterpart of the $q$-Schur superalgebra (of type A) formulated
as centralizer algebras of certain signed $q$-permutation modules over Hecke
algebras. Some multiplication formulas for $\imath$Schur superalgebra are
obtained to construct their canonical bases. Furthermore, we established an
isomorphism theorem between the $\imath$Scuhr superalgebras and the $q$-Schur
superalgebras of type A, which helps us derive a semisimplicity criteria of the
$\imath$Schur superalgebras.
