On the class groups of certain imaginary cyclic fields of 2-power degree

Let p be an odd prime number and let 2(e+1) be the highest power of 2 dividing p - 1. For 0 <= n <= e, let k(n) be the real cyclic field of conductor p and degree 2(n). For a certain imaginary quadratic field L-0, we put L-n = L(0)k(n). For 0 <= n <= e - 1, let F-n be the imaginary quadratic subextension of the imaginary (2,2)-extension Ln+1/k(n), with F-n not equal L-n. We study the Galois module structure of the 2-part of the ideal class group of the imaginary cyclic field F-n. This generalizes a classical result of Rdei and Reichardt for the case n = 0.