The computation of $\mathrm{Gal}(k^\infty /k)$ for some complex quadratic number fields $k$

We determine $\mathrm{Gal}(k^\infty /k)$ up to isomorphism for two particular families of imaginary quadratic number fields $k$ with $2$-class field tower of length $2$, where $k^\infty $ denotes the union of the $2$-class field tower of $k$.