Decomposition into irreducible of a representation of the wreath product $S_d \wr S_n$ (4)

I have to decompose some representations of $$S_d \wr S_n$$. I understand better and better how it works, I still have a case I don't know how to deal with.

For simplicity I take $$d=2$$ and $$n=4$$. $$S^{(2)}$$ is the trivial irrep and $$S^{(1,1)}$$ the sign irrep of $$S_2$$.

Let us consider the irrep of $$S_2 \wr S_4$$: $$\bigl( U \boxtimes V \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}$$ where $$U=S^{(1,1)}$$ and $$V={S^{(2)}}^{\widetilde{\otimes 3}}$$

and the other irrep $$\bigl( U \boxtimes (V\otimes S^{(2,1)}) \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}$$

I'd like to decompose the product of the two, I don't know how this works...

Thanks a lot!

PS:

$$... = \bigl({S^{(1,1)}}^{\otimes 2}\boxtimes {S^{(2)}}^{\widetilde{\otimes 3}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}\oplus \bigl({S^{(1,1)}}\boxtimes{S^{(1,1)}}\boxtimes {S^{(2)}}^{\widetilde{\otimes 2}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{1}\times S_2\wr S_{2}}^{S_2\wr S_4}$$

With $${S^{(1,1)}}^{\otimes 2}=S^{(2)}$$, the left term is $$\bigl({S^{(2)}}\boxtimes {S^{(2)}}^{\widetilde{\otimes 3}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}= S^{(4)} \oplus S^{(3,1)}$$

The right term is $$\bigl( X \boxtimes Y \bigr)\bigl\uparrow_{S_2\wr S_{2} \times S_2\wr S_{2}}^{S_2\wr S_4} \oplus \bigl( (X\otimes S^{(1,1)}) \boxtimes Y \bigr)\bigl\uparrow_{S_2\wr S_{2} \times S_2\wr S_{2}}^{S_2\wr S_4}$$, where $$X={S^{(1,1)}}^{\widetilde{\otimes 2}}$$ and $$Y={S^{(2)}}^{\widetilde{\otimes 2}}$$.

Is this correct? Should I write this differently? I suspect my "$$\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}$$" to be unappropriate.