Apologies if this question is really silly, but this example only makes me more confused, because I know $\mathrm{End}_{\mathbb{C}[S_3]}(V \oplus V) = \mathrm{End}_{\mathbb{C}[S_3]}(V\otimes V^\ast) = \mathrm{End}_{\mathbb{C}}(V^\ast) = M_2(\mathbb C)$. But if this argument is formal, then I think $\mathrm{End}(\mathbb {C}[S_3]) = \mathbb{C}[S_3]$ and $\mathbb{C}[S_3] = \bigoplus_{\lambda\in \mathrm{Irr}(S_3)} V_{\lambda} \otimes V^\ast_{\lambda}$ should imply $\mathrm{End}(V^\ast_{\lambda}) = \mathbb{C}$...

Thank you for your comment. One thing I still do not understand is how the endomorphism ring can be just Q if the multiplicity is larger than one. This is what I was trying to say with the last paragraph.