Let's take $M$ to be $Z/3$ with trivial $S _3$ action for concreteness' sake. Then we have $$H^*(C_3,M)\cong Z/3[x_2]\otimes \Lambda (e_1).$$ Now, $S_3/C_3\cong F_3 ^*$ acts on this ring just by multiplication by constant on $x_2$. So as $a^2=1$ for all $a \in F_3 ^*$ by small Fermat's theorem, invariants are $ Z/3[x_2^2]\otimes \Lambda (e_1).$ Thus we pass from period 2 to period 4.

Let's take $M$ to be $Z/3$ with trivial $S _3$ action for concreteness' sake. Then we have $$H^*(C_3,M)\cong Z/3[x_2]\otimes \Lambda (e_1).$$ Now, $S_3/C_3\cong F_3 ^*$ acts on this ring just by multiplication by constant on $x_2$. So as $a^2=1$ for all $a \in F_3 ^*$ by Fermat's small theorem, invariants are $ Z/3[x_2^2]\otimes \Lambda (e_1).$ Thus we pass from period 2 to period 4. In other words, the $S_3$ action differs on dim $4n$ and $4n+2$.