# gr.group theory – Group cohomology with coefficients in a graded module

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 $$H^*(G,M^*)$$.