  Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic
fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic
field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and
k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the
possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M)
of the second Hilbert 5-class field M(5,2) of M are investigated, when the
5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal
unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already.
The precise length ell5(M) of the 5-class tower of M is determined, when
G(5,2)M is of order less than or equal to 5^5. Theoretical results are
underpinned by the actual computation of all 83, respectively 93, cases in the
range 0<d<10^4, respectively -2*10^5<d<0.
