1.6.92 Fields, Groups, Rings

Condition: Abeleva Group \ (A \) contains as a subgroup \ (\ mathb {z} / 2 p \ mathbbbbbbbbbb {z}, \ quad \) and the factor-group is isomorphic \ (\ mathbb {z} / 2 Q {z} \), moreover \ (p, q \) - simple. What can be isomorphic group \ (A \)? Consider all the remaining cases.