Any $${\mathcal {S}} \in \mathfrak {sp}(1,{\mathbb {R}})$$ induces canonically a derivation S of the Heisenberg Lie algebra $${\mathfrak {h}}$$ and so, a semi-direct extension $$G_{{\mathcal {S}}}=H \rtimes \exp ({\mathbb {R}}S)$$ of the Heisenberg Lie group H (Muller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group $$G_{{\mathcal {S}}}$$ and a family $$g_a$$ of left-invariant (Lorentzian and Riemannian) metrics on $$G_{{\mathcal {S}}}$$ , which generalize the case of the oscillator group. Both the Lie algebra and the analytic description will be used to investigate the geometry of $$(G_{{\mathcal {S}}},g_a)$$ , with particular regard to the study of nontrivial Ricci solitons.