= Transgression in group cohomology
For a <group extension> $1\to H\to G\to Q\to1$ and a $G$-<module> $M$, the transgression
$$
d_2:H^1(H,M)^Q\longrightarrow H^2(Q,M^H)
$$
is the first <spectral sequence differential> crossing from the vertical edge to the horizontal edge of the <Lyndon–Hochschild–Serre spectral sequence>. By the <five-term exact sequence in group cohomology>, an invariant <one-cocycle> class on $H$ has zero transgression precisely when it is the restriction of a <one-cocycle> class on $G$. Thus transgression measures an extension obstruction. An arbitrary extension of a representative to a <group cochain> on $G$ need not have a coboundary that already descends to $Q$; the cochain construction requires appropriate corrections.
For $1\to C_2\to C_4\to C_2\to1$ with trivial coefficients $\mathbb F_2$, every <group homomorphism> $C_4\to\mathbb F_2$ kills its order-two <subgroup>. Consequently the nonzero class in $H^1(C_2,\mathbb F_2)$ has nonzero transgression. It detects this nonsplit <group extension>, represented by the carry <two-cocycle> $c(a,b)=ab$ for $a,b\in\{0,1\}$.
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