On a composition of subfactors with group subfactors (Q829454)

The authors study the lattice of intermediate subfactors of a subfactor, which is an important topic in subfactor theory. The paper has interesting results and it is well written. The authors construct a (finite-depth) subfactor \(N \subseteq M\) from a composition of a (finite-depth) subfactor \(N\subseteq P\) and a group subfactor \(P\subset M\). The lattice of intermediate subfactors of \(N \subseteq M\) is a free composition of the lattices of intermediate subfactors \(N\subseteq P\) and \(P\subset M\). \textit{D. Bisch} and \textit{U. Haagerup} [unpublished manuscript (1994)] studied such a composition for the \(Z_2\) subfactor and the \(A_4\) subfactor as ``the second fish''. Furthermore, the authors ask the question ``Is it possible to realize the composed lattice of two finite depth intermediate subfactor lattices as an intermediate subfactor lattice for some irreducible subfactor \(N \subseteq M\) with finite index and finite depth?'', which is true without the finite depth requirement, as constructed by the reviewer [Trans. Am. Math. Soc. 368, No. 12, 8303--8348 (2016; Zbl 1365.46053)] using Bisch-Jones free composition of subfactors.