For any subgroup $H$ of a fnite group $G$ and any $x ∈ G$, the right $kH$-module $k[xH]$ is also a left $k^{x}H$-module because $xH = xHx^{−1}x = {^xH}x$ is also a left $^xH$-coset. Thus $k[xH]$ is in fact a $k^xH-kH$-bimodule, and for any $kH$-module $V$ we have a $k^xH$-module $^xV = k[xH] \otimes_{kH} V$.
Equivalently, we could have defned $^xV$ by setting $^xV = V$ as a $k$-module, with $h∈{^x}H$ acting on $v∈ {^x}V$ as $x^{−1}hx$ on $v ∈ V$.
So for $N$ a normal subgroup of $G$, $S$ a simple $kN$-module, $^xS$ is a simple $kN$-module for any $x ∈ G$. How come there are cases where $S \ncong {^xS}$ for all $x \in G$ \ $N$. I think they are all isomorphic. Any help would be appreciated!
