Representation theory – decomposition of a representation of the wreath product into irreducible $ S_d wr S_n $ (3)

To let:
$$ R_m ^ n = bigl (F ^ { widetilde { otimes nm}} boxtimes S ^ { widetilde { otimes m}} bigr) bigl uparrow_ {S_ {nm} times S_ {m }} ^ {S_m}: $$
This is a non-reducible representation of $ S_d wr S_n $,

I would like to know how to disassemble the Kronecker product $ R_m ^ n is sometimes R_n ^ n $, More Genenraly, it is easy to disassemble $ R_m ^ n is sometimes R_n ^ n $ in irreducible representations?

Many Thanks!

(related question: //mathoverflow.net/questions/317485/en-composition-unseparable-of-the-the-the-granzproduct-s-d-wr)