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$B\rtimes\Sigma_n$ is Morita equivalent to $A\rtimes\Sigma_{n-1}$ with $\Sigma_{n-1}$ acting trivially. The latter is isomorphic to $A\otimes \mathbb{C}\Sigma_{n-1}$, which shows that $B\rtimes\Sigma_n$ is not simple. Let $c_n$ be the number of conjugacy classes of $\Sigma_n$. Then $\mathbb{C}\Sigma_{n-1}$ is the direct sum of $c_{n-1}$ matrix algebras, so by invariance of $K_1$ under Morita invariance we have $K_1(B\rtimes\Sigma_n)=\mathbb{Q}^{c_{n-1}}$