A correspondence induced by an involution centralizing an index-two subgroup

Given groups (G, ○) and (H, ⋆) with disjoint support, we remark that the isomorphisms (G, ○) → (H, ⋆) are in one-to-one correspondence with certain group operations on G ∪H extending the operation of G and interacting nicely with the structure on H. This correspondence follows from some general properties of groups admitting an index-two subgroup N with an involution x ~∈ N which centralizes N.