On the rank of the $$2$$2-class group of $$\mathbb {Q}(\sqrt{p}, \sqrt{q},\sqrt{-1} )$$Q(p,q,-1)

Let $$d$$d be a square-free integer, $$\mathbf {k}=\mathbb {Q}(\sqrt{d},\,i)$$k=Q(d,i) and $$i=\sqrt{-1}$$i=-1. Let $$\mathbf {k}_1^{(2)}$$k1(2) be the Hilbert $$2$$2-class field of $$\mathbf {k}$$k, $$\mathbf {k}_2^{(2)}$$k2(2) be the Hilbert $$2$$2-class field of $$\mathbf {k}_1^{(2)}$$k1(2) and $$G=\mathrm {Gal}(\mathbf {k}_2^{(2)}/\mathbf {k})$$G=Gal(k2(2)/k) be the Galois group of $$\mathbf {k}_2^{(2)}/\mathbf {k}$$k2(2)/k. We give necessary and sufficient conditions to have $$G$$G metacyclic in the case where $$d=pq$$d=pq, with $$p$$p and $$q$$q primes such that $$p\equiv 1\pmod 8$$p≡1(mod8) and $$q\equiv 5\pmod 8$$q≡5(mod8) or $$p\equiv 1\pmod 8$$p≡1(mod8) and $$q\equiv 3\pmod 4$$q≡3(mod4).