Let $0 \to A \to X \to C \to 0$ be an abelian group extension (we require $X$ to be abelian too). Then the group operation on $X$ is described by a $2$-cocycle $c(x, y) = s(x) + s(y) -s(x+y)$ where $s: C \to X $ is a set-theoretic section of the epimorphism from $X$ to $C$. The sequence splits if and only if there is a section $s$ which is also an homomorphism, so that the associated $c \equiv 0$.
My question is: what kind of extensions do we get in case $c $ is bilinear? This should be equivalent to linearity on just one component as $c: C \times C \to X$ should be symmetric if I recall correctly. This is clearly satisfied if the sequence splits. Are there any nontrivial cases?