I am working in a problem group cohomology and nailed it down to compute a cohomology using an spectral sequence argument. The situation is as follows:

Let $G = C_4 = langle sigma rangle$ be the cyclic group of order $4$, $k = mathbb{F}_2$ and $M^*=M^0 oplus M^1 oplus M^2$ be a graded module where $M^1 = M^2 = k$ is the trivial $G$-module and $M^1 = k oplus k$ be the $G$-module where $sigma(a,b) = (b,a)$.

To compute $H^*(G, M^*)$, we can use a spectral sequence argument with $E_2^{p,q} =H^p(G,M^q)$. Since $M^0$ and $M^2$ are trivial, $E_2^{p,q} = H^p(G,k) otimes M^q$ for $q=0,2$.

When $q = 1$, we can use Shapiro’s lemma to show that $E_2^{p,1} = H^p(C_2;k) cong k(t)$ where $C_2 = langle sigma^2 rangle$.

However, I have no leads on how to compute the differential $d_2:E_2^{p,1} = H^p(C_2,k) rightarrow E_2^{p+2,0} = H^{p+2}(C_4,k)$.

I appreciate any input in this computation, or if there is any alternative to describe $H^*(G,M^*)$.