Extensions with normal subgroup Q8 and quotient group Z2

This article describes the full solution to the group extension problem for normal subgroup the quaternion group (of order eight) and quotient group cyclic group:Z2: find all congruence classes of group extensions with normal subgroup identified with the quaternion group and quotient group identified with cyclic group:Z2.

We follow the procedure outlined at the group extension problem page, so we set $$N$$ to be the quaternion group and $$Q$$ to be cyclic group:Z2.