The author determines all non-quadratic imaginary cyclic number fields \(K\) of 2-power degree whose ideal class groups have exponents less than or equal to 2. It is shown that there are 38 such fields, 33 of degree 4, 4 of degree 8, and 1 of degree 16. There are several references in the bibliography, which are not referred to in the text. Insufficient acknowledgement of the work of earlier authors is made (for example the cases \(h=1\) and \(h=2\) of Theorem 13).