# Second cohomology group for trivial group action of Q8 on V4

This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group quaternion group on Klein four-group. The elements of this classify the group extensions with Klein four-group in the center and quaternion group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
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## Description of the group

$\! H^2(G;A)$
where $G$ is the quaternion group (of order 8) and $A$ is the Klein four-group $V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.