I'm trying to understand the representations of $k\mathfrak{S}_3$, where $k$ is a field of characteristic $2$.
Could someone explain the permutation module $M^{(1^3)}$ has a filtration layers isomorphic to $S^{(1^3)}, 2S^{(2,1)}$ and $S^{(3)}$?
Note that $S^\lambda$ is the Specht module corresponding to the partition $\lambda$.
Moreover, how can we use Frobenius reciprocity to conclude that $M^{(1^3)}$ contains one copy of the trivial module $k$ in its head and socle?
Thank you!